Understanding and Operating with Polynomials - Algebra 2
Card 1 of 30
Identify the polynomial degree: $x^7+3x^2-8x+1$.
Identify the polynomial degree: $x^7+3x^2-8x+1$.
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$7$. Highest exponent determines the degree.
$7$. Highest exponent determines the degree.
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What is the product $(x^2-4)(x+1)$?
What is the product $(x^2-4)(x+1)$?
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$x^3+x^2-4x-4$. Distribute first polynomial to each term in second.
$x^3+x^2-4x-4$. Distribute first polynomial to each term in second.
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What is the product $(x^2+2x)(x-3)$?
What is the product $(x^2+2x)(x-3)$?
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$x^3-x^2-6x$. Distribute first polynomial to each term in second.
$x^3-x^2-6x$. Distribute first polynomial to each term in second.
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What is the product $(2x+1)(x^2-x+3)$?
What is the product $(2x+1)(x^2-x+3)$?
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$2x^3-x^2+5x+3$. Distribute binomial to each term in the polynomial.
$2x^3-x^2+5x+3$. Distribute binomial to each term in the polynomial.
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What is the product $-2x^2(3x^3-x+6)$?
What is the product $-2x^2(3x^3-x+6)$?
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$-6x^5+2x^3-12x^2$. Distribute the monomial to each term.
$-6x^5+2x^3-12x^2$. Distribute the monomial to each term.
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What is the product $(x-4)(x+6)$?
What is the product $(x-4)(x+6)$?
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$x^2+2x-24$. Use FOIL method to multiply binomials.
$x^2+2x-24$. Use FOIL method to multiply binomials.
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What is the product $(x-1)(x^2+2x+3)$?
What is the product $(x-1)(x^2+2x+3)$?
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$x^3+x^2+x-3$. Distribute binomial to each term in the polynomial.
$x^3+x^2+x-3$. Distribute binomial to each term in the polynomial.
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What is the product $(x+2)(x^2+3x+4)$?
What is the product $(x+2)(x^2+3x+4)$?
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$x^3+5x^2+10x+8$. Distribute binomial to each term in the polynomial.
$x^3+5x^2+10x+8$. Distribute binomial to each term in the polynomial.
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What is the product $3x(2x^2-5x+4)$?
What is the product $3x(2x^2-5x+4)$?
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$6x^3-15x^2+12x$. Distribute the monomial to each term.
$6x^3-15x^2+12x$. Distribute the monomial to each term.
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What is the product $(x+3)^2$ expanded?
What is the product $(x+3)^2$ expanded?
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$x^2+6x+9$. Square the binomial: $(a+b)^2=a^2+2ab+b^2$.
$x^2+6x+9$. Square the binomial: $(a+b)^2=a^2+2ab+b^2$.
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What is the result of combining like terms: $-x^3+4x^2+7-x^3-2x^2$?
What is the result of combining like terms: $-x^3+4x^2+7-x^3-2x^2$?
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$-2x^3+2x^2+7$. Group and add coefficients of like terms.
$-2x^3+2x^2+7$. Group and add coefficients of like terms.
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What is the result of combining like terms: $5x^2-3x+2x^2+9x-4$?
What is the result of combining like terms: $5x^2-3x+2x^2+9x-4$?
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$7x^2+6x-4$. Group and add coefficients of like terms.
$7x^2+6x-4$. Group and add coefficients of like terms.
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What is the result of $-(3x^2-4x+6)$ written as a polynomial?
What is the result of $-(3x^2-4x+6)$ written as a polynomial?
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$-3x^2+4x-6$. Distribute the negative sign to all terms.
$-3x^2+4x-6$. Distribute the negative sign to all terms.
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What is the product $(x-2)^2$ expanded?
What is the product $(x-2)^2$ expanded?
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$x^2-4x+4$. Square the binomial: $(a-b)^2=a^2-2ab+b^2$.
$x^2-4x+4$. Square the binomial: $(a-b)^2=a^2-2ab+b^2$.
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What is the result of subtracting $ (2x^3+x-9) - (5x^3-4x+1)$?
What is the result of subtracting $ (2x^3+x-9) - (5x^3-4x+1)$?
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$-3x^3+5x-10$. Distribute the negative and combine like terms.
$-3x^3+5x-10$. Distribute the negative and combine like terms.
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What is the result of subtracting $ (4x^2-6x+1) - (x^2+2x-3)$?
What is the result of subtracting $ (4x^2-6x+1) - (x^2+2x-3)$?
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$3x^2-8x+4$. Distribute the negative and combine like terms.
$3x^2-8x+4$. Distribute the negative and combine like terms.
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What is the result of adding $ (7x^3-2x) + (-7x^3+9x+5)$?
What is the result of adding $ (7x^3-2x) + (-7x^3+9x+5)$?
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$7x+5$. The cubic terms cancel, leaving linear and constant.
$7x+5$. The cubic terms cancel, leaving linear and constant.
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What is the result of adding $ (3x^2+5x-1) + (2x^2-3x+4)$?
What is the result of adding $ (3x^2+5x-1) + (2x^2-3x+4)$?
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$5x^2+2x+3$. Add coefficients of like terms separately.
$5x^2+2x+3$. Add coefficients of like terms separately.
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What is the product $(x^2+1)(x^2-1)$?
What is the product $(x^2+1)(x^2-1)$?
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$x^4-1$. Use difference of squares formula: $a^2-b^2$.
$x^4-1$. Use difference of squares formula: $a^2-b^2$.
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What is the degree of the polynomial $6x^2-4x+9$?
What is the degree of the polynomial $6x^2-4x+9$?
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$2$. Highest exponent determines the degree.
$2$. Highest exponent determines the degree.
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What is the leading coefficient of the polynomial $-3x^4+5x^2-1$?
What is the leading coefficient of the polynomial $-3x^4+5x^2-1$?
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$-3$. Coefficient of the highest degree term.
$-3$. Coefficient of the highest degree term.
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What is the leading term of the polynomial $4x^3-2x+7$?
What is the leading term of the polynomial $4x^3-2x+7$?
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$4x^3$. The term with the highest degree comes first.
$4x^3$. The term with the highest degree comes first.
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What is the degree of the constant monomial $-9$?
What is the degree of the constant monomial $-9$?
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$0$. Constants have degree zero by definition.
$0$. Constants have degree zero by definition.
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What is the degree of the monomial $7x^5$?
What is the degree of the monomial $7x^5$?
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$5$. The degree is the exponent of the variable.
$5$. The degree is the exponent of the variable.
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What is a like term in a polynomial expression?
What is a like term in a polynomial expression?
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A term with the same variable part, such as $x^3$ terms. Same variables raised to the same powers.
A term with the same variable part, such as $x^3$ terms. Same variables raised to the same powers.
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What is the standard form of a polynomial written in one variable $x$?
What is the standard form of a polynomial written in one variable $x$?
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Terms in descending powers of $x$ with like terms combined. Highest to lowest powers with simplified terms.
Terms in descending powers of $x$ with like terms combined. Highest to lowest powers with simplified terms.
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What does it mean that polynomials are closed under multiplication?
What does it mean that polynomials are closed under multiplication?
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The product of two polynomials is always a polynomial. The result stays within the polynomial system.
The product of two polynomials is always a polynomial. The result stays within the polynomial system.
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What does it mean that polynomials are closed under subtraction?
What does it mean that polynomials are closed under subtraction?
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The difference of two polynomials is always a polynomial. The result stays within the polynomial system.
The difference of two polynomials is always a polynomial. The result stays within the polynomial system.
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What does it mean that polynomials are closed under addition?
What does it mean that polynomials are closed under addition?
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The sum of two polynomials is always a polynomial. The result stays within the polynomial system.
The sum of two polynomials is always a polynomial. The result stays within the polynomial system.
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Find the product $(x+1)(x^2-1)$ in standard form.
Find the product $(x+1)(x^2-1)$ in standard form.
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$x^3+x^2-x-1$. Expand $(x+1)(x+1)(x-1)$ step by step.
$x^3+x^2-x-1$. Expand $(x+1)(x+1)(x-1)$ step by step.
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