Multiply Factored Expressions - International Brotherhood of Electrical Workers Exam IBEW

In analyzing a circuit, an expression $(I + 3)(2I - 1)$ arises. What is this expression when expanded?
Answer: $2I^2 + 5I - 3$ When you encounter algebraic expressions in electrical circuit analysis, you'll often need to expand products of binomials. This tests your ability to apply the distributive property systematically. To expand $(I + 3)(2I - 1)$ , you multiply each term in the first binomial by each term in the second binomial. Using the FOIL method: First terms: $I \times 2I = 2I^2$ . Outer terms: $I \times (-1) = -I$ . Inner terms: $3 \times 2I = 6I$ . Last terms: $3 \times (-1) = -3$ . Combining these gives: $2I^2 - I + 6I - 3 = 2I^2 + 5I - 3$ . Answer choice A ( $3I + 2$ ) completely ignores the quadratic nature of multiplying two linear expressions and appears to result from incorrectly adding coefficients rather than multiplying terms. Answer choice B ( $2I^2 - 5I - 3$ ) gets the quadratic and constant terms correct but has the wrong sign on the linear term—this suggests combining $-I + 6I$ as $-5I$ instead of $+5I$ . Answer choice C ( $2I^2 + 7I - 3$ ) correctly identifies the quadratic and constant terms but miscalculates the middle term, possibly by adding $1 + 6 = 7$ instead of properly handling $-1 + 6 = 5$ . The key strategy is to be methodical with FOIL and carefully track positive and negative signs when combining like terms. Double-check your arithmetic on the middle term, as sign errors there are the most common mistake in binomial multiplication.
Expand $(3x + 6)(x + 1)$ using distribution.
Answer: $3x^2 + 9x + 6$ This question tests the ability to identify and expand products from factored algebraic expressions, a key skill in algebra and functions. The process involves applying the distributive property, multiplying each term in the first polynomial by each term in the second, then combining like terms. For example, in the expression (3x + 6)(x + 1), each term in the first binomial is multiplied by each term in the second, resulting in $3x^2$ + 3x + 6x + 6. The correct answer is ' $3x^2$ + 9x + 6' because combining the like terms 3x and 6x gives 9x. A common error is failing to multiply each term correctly, leading to incorrect answers such as ' $3x^2$ + 7x + 6'. Teaching strategies include practicing the distributive property with various expressions, using visual aids like area models to illustrate multiplication, and reinforcing the combination of like terms.
Expand $(6x + 5)(x + 2)$ using distributive multiplication.
Answer: $6x^2 + 17x + 10$ This question tests the ability to identify and expand products from factored algebraic expressions, a key skill in algebra and functions. The process involves applying the distributive property, multiplying each term in the first polynomial by each term in the second, then combining like terms. For example, in the expression (6x + 5)(x + 2), each term in the first binomial is multiplied by each term in the second, resulting in $6x^2$ + 12x + 5x + 10. The correct answer is ' $6x^2$ + 17x + 10' because combining the like terms 12x and 5x gives 17x. A common error is failing to multiply each term correctly, leading to incorrect answers such as ' $6x^2$ + 12x + 10'. Teaching strategies include practicing the distributive property with various expressions, using visual aids like area models to illustrate multiplication, and reinforcing the combination of like terms.
Find the expanded form of $(3x + 2)(x + 7)$ .
Answer: $3x^2 + 23x + 14$ This question tests the ability to identify and expand products from factored algebraic expressions, a key skill in algebra and functions. The process involves applying the distributive property, multiplying each term in the first polynomial by each term in the second, then combining like terms. For example, in the expression (3x + 2)(x + 7), each term in the first binomial is multiplied by each term in the second, resulting in $3x^2$ + 21x + 2x + 14. The correct answer is ' $3x^2$ + 23x + 14' because combining the like terms 21x and 2x gives 23x. A common error is failing to multiply each term correctly, leading to incorrect answers such as ' $3x^2$ + 21x + 14'. Teaching strategies include practicing the distributive property with various expressions, using visual aids like area models to illustrate multiplication, and reinforcing the combination of like terms.
Multiply the expression $(x + \frac{1}{2})(x - \frac{1}{2})$ .
Answer: $x^2 - \frac{1}{4}$ When you encounter expressions like $(x + \frac{1}{2})(x - \frac{1}{2})$ , recognize this as a difference of squares pattern. This special form $(a + b)(a - b)$ always equals $a^2 - b^2$ , which is much faster than using FOIL. Here, $a = x$ and $b = \frac{1}{2}$ , so the result is $x^2 - (\frac{1}{2})^2 = x^2 - \frac{1}{4}$ . Let's verify with FOIL: First terms give $x \cdot x = x^2$ , Outer terms give $x \cdot (-\frac{1}{2}) = -\frac{x}{2}$ , Inner terms give $\frac{1}{2} \cdot x = \frac{x}{2}$ , and Last terms give $\frac{1}{2} \cdot (-\frac{1}{2}) = -\frac{1}{4}$ . The middle terms cancel: $x^2 - \frac{x}{2} + \frac{x}{2} - \frac{1}{4} = x^2 - \frac{1}{4}$ , confirming answer D. Answer A ( $x^2 - x - \frac{1}{4}$ ) incorrectly keeps the middle terms instead of recognizing they cancel. Answer B ( $x^2 + \frac{1}{4}$ ) makes a sign error when squaring $-\frac{1}{2}$ , giving a positive result instead of negative. Answer C ( $2x$ ) completely misapplies distribution rules and ignores the quadratic nature of the multiplication. Remember the difference of squares shortcut: $(a + b)(a - b) = a^2 - b^2$ . On electrical exams, this pattern appears frequently in impedance calculations and power formulas, so mastering it saves valuable time during the test.
Multiply $(x + 8)(x + 1)$ and simplify the result.
Answer: $x^2 + 9x + 8$ This question tests the ability to identify and expand products from factored algebraic expressions, a key skill in algebra and functions. The process involves applying the distributive property, multiplying each term in the first polynomial by each term in the second, then combining like terms. For example, in the expression (x + 8)(x + 1), each term in the first binomial is multiplied by each term in the second, resulting in $x^2$ + x + 8x + 8. The correct answer is ' $x^2$ + 9x + 8' because combining the like terms x and 8x gives 9x. A common error is failing to multiply each term correctly, leading to incorrect answers such as ' $x^2$ + 8x + 1'. Teaching strategies include practicing the distributive property with various expressions, using visual aids like area models to illustrate multiplication, and reinforcing the combination of like terms.
Expand the product $(x + 6)(x + 6)$ into standard form.
Answer: $x^2 + 12x + 36$ This question tests the ability to identify and expand products from factored algebraic expressions, a key skill in algebra and functions. The process involves applying the distributive property, multiplying each term in the first polynomial by each term in the second, then combining like terms. For example, in the expression (x + 6)(x + 6), each term in the first binomial is multiplied by each term in the second, resulting in $x^2$ + 6x + 6x + 36. The correct answer is ' $x^2$ + 12x + 36' because combining the like terms 6x and 6x gives 12x. A common error is failing to multiply each term correctly, leading to incorrect answers such as ' $x^2$ + 6x + 36'. Teaching strategies include practicing the distributive property with various expressions, using visual aids like area models to illustrate multiplication, and reinforcing the combination of like terms.
Expand $(2x + 3)(4x + 1)$ and simplify completely.
Answer: $8x^2 + 14x + 3$ This question tests the ability to identify and expand products from factored algebraic expressions, a key skill in algebra and functions. The process involves applying the distributive property, multiplying each term in the first polynomial by each term in the second, then combining like terms. For example, in the expression (2x + 3)(4x + 1), each term in the first binomial is multiplied by each term in the second, resulting in $8x^2$ + 2x + 12x + 3. The correct answer is ' $8x^2$ + 14x + 3' because combining the like terms 2x and 12x gives 14x. A common error is failing to multiply each term correctly, leading to incorrect answers such as ' $8x^2$ + 12x + 3'. Teaching strategies include practicing the distributive property with various expressions, using visual aids like area models to illustrate multiplication, and reinforcing the combination of like terms.
Expand $(2x + 1)(2x + 5)$ and combine like terms.
Answer: $4x^2 + 12x + 5$ This question tests the ability to identify and expand products from factored algebraic expressions, a key skill in algebra and functions. The process involves applying the distributive property, multiplying each term in the first polynomial by each term in the second, then combining like terms. For example, in the expression (2x + 1)(2x + 5), each term in the first binomial is multiplied by each term in the second, resulting in $4x^2$ + 10x + 2x + 5. The correct answer is ' $4x^2$ + 12x + 5' because combining the like terms 10x and 2x gives 12x. A common error is failing to multiply each term correctly, leading to incorrect answers such as ' $4x^2$ + 10x + 5'. Teaching strategies include practicing the distributive property with various expressions, using visual aids like area models to illustrate multiplication, and reinforcing the combination of like terms.
An electrical signal's voltage is modeled by $(t + 0.5)(t - 0.2)$ . What is the expanded form?
Answer: $t^2 + 0.3t - 0.1$ This question tests your ability to expand binomial expressions using the FOIL method (First, Outer, Inner, Last), a fundamental algebraic skill needed for analyzing electrical circuit equations and signal modeling. To expand $(t + 0.5)(t - 0.2)$ , multiply each term in the first binomial by each term in the second: First terms: $t \times t = t^2$ Outer terms: $t \times (-0.2) = -0.2t$ Inner terms: $0.5 \times t = 0.5t$ Last terms: $0.5 \times (-0.2) = -0.1$ Combining all terms: $t^2 - 0.2t + 0.5t - 0.1$ Combining like terms: $t^2 + 0.3t - 0.1$ This confirms answer choice D is correct. Let's examine why the other options are wrong: Choice A ( $t^2 - 0.3t - 0.1$ ) incorrectly shows the middle term as negative. This happens if you forget that $-0.2t + 0.5t = +0.3t$ , not $-0.3t$ . Choice B ( $t^2 + 0.7t - 0.1$ ) has the wrong coefficient for the middle term. This error occurs if you add the coefficients incorrectly: $0.5 + 0.2 = 0.7$ instead of $0.5 + (-0.2) = 0.3$ . Choice C ( $t^2 - 0.1$ ) is missing the middle term entirely, suggesting the student ignored the outer and inner products in FOIL. Study tip: When expanding binomials, always write out all four FOIL products before combining like terms. This prevents sign errors and ensures you don't miss any terms.
What is the product of $(3x + 5y)(3x - 5y)$ ?
Answer: $9x^2 - 25y^2$ This question tests your ability to recognize and apply the difference of squares pattern, a fundamental algebraic concept that appears frequently on technical exams. When you see two binomials in the form $(a + b)(a - b)$ , you're looking at a difference of squares. This special pattern always multiplies to $a^2 - b^2$ . Here, $a = 3x$ and $b = 5y$ , so the product is $(3x)^2 - (5y)^2 = 9x^2 - 25y^2$ . You can verify this using FOIL: First terms give $3x \cdot 3x = 9x^2$ , Outer terms give $3x \cdot (-5y) = -15xy$ , Inner terms give $5y \cdot 3x = 15xy$ , and Last terms give $5y \cdot (-5y) = -25y^2$ . Notice the middle terms $-15xy + 15xy = 0$ , leaving only $9x^2 - 25y^2$ . Choice A ( $6x$ ) completely ignores the $y$ terms and incorrectly adds coefficients instead of multiplying. Choice B ( $9x^2 + 25y^2$ ) correctly squares each term but uses addition instead of subtraction—this would be the sum of squares, not difference. Choice C ( $9x^2 - 30xy - 25y^2$ ) represents a common error where students incorrectly keep the middle terms instead of recognizing they cancel out. Study tip: Memorize the difference of squares pattern: $(a + b)(a - b) = a^2 - b^2$ . When you see this structure, the middle terms always cancel, leaving only the squared terms with subtraction. This saves time and prevents calculation errors on electrical exams where algebraic manipulation is essential.
Find the expanded polynomial for $(7x + 1)(x + 8)$ .
Answer: $7x^2 + 57x + 8$ This question tests the ability to identify and expand products from factored algebraic expressions, a key skill in algebra and functions. The process involves applying the distributive property, multiplying each term in the first polynomial by each term in the second, then combining like terms. For example, in the expression (7x + 1)(x + 8), each term in the first binomial is multiplied by each term in the second, resulting in $7x^2$ + 56x + x + 8. The correct answer is ' $7x^2$ + 57x + 8' because combining the like terms 56x and x gives 57x. A common error is failing to multiply each term correctly, leading to incorrect answers such as ' $7x^2$ + 56x + 8'. Teaching strategies include practicing the distributive property with various expressions, using visual aids like area models to illustrate multiplication, and reinforcing the combination of like terms.
Expand $(4x + 7)(2x + 3)$ into a quadratic polynomial.
Answer: $8x^2 + 26x + 21$ This question tests the ability to identify and expand products from factored algebraic expressions, a key skill in algebra and functions. The process involves applying the distributive property, multiplying each term in the first polynomial by each term in the second, then combining like terms. For example, in the expression (4x + 7)(2x + 3), each term in the first binomial is multiplied by each term in the second, resulting in $8x^2$ + 12x + 14x + 21. The correct answer is ' $8x^2$ + 26x + 21' because combining the like terms 12x and 14x gives 26x. A common error is failing to multiply each term correctly, leading to incorrect answers such as ' $8x^2$ + 20x + 21'. Teaching strategies include practicing the distributive property with various expressions, using visual aids like area models to illustrate multiplication, and reinforcing the combination of like terms.
Expand $(2x + 7)(x + 3)$ and combine like terms.
Answer: $2x^2 + 13x + 21$ This question tests the ability to identify and expand products from factored algebraic expressions, a key skill in algebra and functions. The process involves applying the distributive property, multiplying each term in the first polynomial by each term in the second, then combining like terms. For example, in the expression (2x + 7)(x + 3), each term in the first binomial is multiplied by each term in the second, resulting in $2x^2$ + 6x + 7x + 21. The correct answer is ' $2x^2$ + 13x + 21' because combining the like terms 6x and 7x gives 13x. A common error is failing to multiply each term correctly, leading to incorrect answers such as ' $2x^2$ + 10x + 21'. Teaching strategies include practicing the distributive property with various expressions, using visual aids like area models to illustrate multiplication, and reinforcing the combination of like terms.
Simplify the product $(x + 9)(x + 2)$ by expansion.
Answer: $x^2 + 11x + 18$ This question tests the ability to identify and expand products from factored algebraic expressions, a key skill in algebra and functions. The process involves applying the distributive property, multiplying each term in the first polynomial by each term in the second, then combining like terms. For example, in the expression (x + 9)(x + 2), each term in the first binomial is multiplied by each term in the second, resulting in $x^2$ + 2x + 9x + 18. The correct answer is ' $x^2$ + 11x + 18' because combining the like terms 2x and 9x gives 11x. A common error is failing to multiply each term correctly, leading to incorrect answers such as ' $x^2$ + 18x + 11'. Teaching strategies include practicing the distributive property with various expressions, using visual aids like area models to illustrate multiplication, and reinforcing the combination of like terms.
Expand and simplify $(5a - 2)(3a - 4)$ .
Answer: $15a^2 - 26a + 8$ When you encounter binomial multiplication problems like this, you're applying the distributive property (also called FOIL) to expand two expressions in parentheses. To expand $(5a - 2)(3a - 4)$ , multiply each term in the first binomial by each term in the second binomial: First terms: $(5a)(3a) = 15a^2$ Outer terms: $(5a)(-4) = -20a$ Inner terms: $(-2)(3a) = -6a$ Last terms: $(-2)(-4) = +8$ Combining these: $15a^2 - 20a - 6a + 8 = 15a^2 - 26a + 8$ This matches answer choice D. Let's examine why the other options are incorrect: Answer A gives $15a^2 + 26a + 8$ . This results from incorrectly making the middle term positive instead of negative, likely from sign errors when multiplying negative terms. Answer B shows $15a^2 - 14a + 8$ . Here, the middle term coefficient is wrong. You'd get -14a if you incorrectly calculated the outer and inner terms as $-20a + 6a$ instead of $-20a - 6a$ . Answer C gives $15a^2 + 8$ , completely missing the middle term. This suggests forgetting to multiply the outer and inner terms entirely. Remember this key strategy: When expanding binomials, be extra careful with negative signs and always double-check that you've multiplied all four term combinations. The most common errors on these problems involve sign mistakes or forgetting terms, so work systematically through each multiplication step.
Simplify the expression $3(x + 2)(x - 1)$ .
Answer: $3x^2 + 3x - 6$ When you encounter algebraic expressions involving multiplication and parentheses, you need to apply the distributive property systematically. This question tests your ability to expand and simplify expressions with multiple factors. To simplify $3(x + 2)(x - 1)$ , first expand the two binomials using FOIL (First, Outer, Inner, Last). Multiply $(x + 2)(x - 1)$ : - First: $x \cdot x = x^2$ - Outer: $x \cdot (-1) = -x$ - Inner: $2 \cdot x = 2x$ - Last: $2 \cdot (-1) = -2$ This gives you $x^2 - x + 2x - 2 = x^2 + x - 2$ . Now multiply this entire result by 3: $3(x^2 + x - 2) = 3x^2 + 3x - 6$ . This matches answer choice A. Answer choice B, $(3x + 6)(3x - 3)$ , represents factoring out 3 from each binomial before expanding, but this changes the mathematical relationship and would give $9x^2 + 9x - 18$ when fully expanded. Answer choice C, $3x^2 + x - 2$ , is what you'd get if you forgot to distribute the 3 to all terms after expanding the binomials. Answer choice D, $3x^2 - 6$ , completely ignores the middle terms that result from the FOIL process. Remember: when expanding expressions with multiple factors, work systematically from inside out. Expand binomials first using FOIL, combine like terms, then apply any remaining multiplication. Double-check by ensuring every term gets multiplied by every factor.
Find the product of $(x + 2y)(x + 3y)$ .
Answer: $x^2 + 5xy + 6y^2$ When you encounter polynomial multiplication like this, you're working with the distributive property, specifically the FOIL method (First, Outer, Inner, Last terms). To find $(x + 2y)(x + 3y)$ , multiply each term in the first binomial by each term in the second: First terms: $x \times x = x^2$ Outer terms: $x \times 3y = 3xy$ Inner terms: $2y \times x = 2xy$ Last terms: $2y \times 3y = 6y^2$ Now combine all terms: $x^2 + 3xy + 2xy + 6y^2$ The middle terms are like terms, so add them: $3xy + 2xy = 5xy$ Final result: $x^2 + 5xy + 6y^2$ Looking at the wrong answers: Choice A ( $x^2 + 6y^2$ ) completely ignores the middle terms that result from multiplying unlike terms together. Choice C ( $x^2 + 6xy + 6y^2$ ) incorrectly combines the middle terms as $6xy$ instead of $5xy$ - this suggests adding $3xy + 3xy$ rather than $3xy + 2xy$ . Choice D ( $x^2 + xy + 6y^2$ ) severely undercounts the middle term, giving only $xy$ instead of $5xy$ . The correct answer is B. Study tip: Always write out all four products when using FOIL, then carefully combine like terms. The coefficient of the middle term should equal the sum of the coefficients from the original binomials (here: $2 + 3 = 5$ ).
Compute the expanded polynomial for $(5x + 1)(x + 4)$ .
Answer: $5x^2 + 21x + 4$ This question tests the ability to identify and expand products from factored algebraic expressions, a key skill in algebra and functions. The process involves applying the distributive property, multiplying each term in the first polynomial by each term in the second, then combining like terms. For example, in the expression (5x + 1)(x + 4), each term in the first binomial is multiplied by each term in the second, resulting in $5x^2$ + 20x + x + 4. The correct answer is ' $5x^2$ + 21x + 4' because combining the like terms 20x and x gives 21x. A common error is failing to multiply each term correctly, leading to incorrect answers such as ' $5x^2$ + 20x + 4'. Teaching strategies include practicing the distributive property with various expressions, using visual aids like area models to illustrate multiplication, and reinforcing the combination of like terms.
Using distribution, expand $(9x + 2)(x + 1)$ correctly.
Answer: $9x^2 + 11x + 2$ This question tests the ability to identify and expand products from factored algebraic expressions, a key skill in algebra and functions. The process involves applying the distributive property, multiplying each term in the first polynomial by each term in the second, then combining like terms. For example, in the expression (9x + 2)(x + 1), each term in the first binomial is multiplied by each term in the second, resulting in $9x^2$ + 9x + 2x + 2. The correct answer is ' $9x^2$ + 11x + 2' because combining the like terms 9x and 2x gives 11x. A common error is failing to multiply each term correctly, leading to incorrect answers such as ' $9x^2$ + 9x + 2'. Teaching strategies include practicing the distributive property with various expressions, using visual aids like area models to illustrate multiplication, and reinforcing the combination of like terms.

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