Understanding and Operating with Polynomials - Algebra
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What is the product $(x-5)(x+2)$ in standard form?
What is the product $(x-5)(x+2)$ in standard form?
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$x^2-3x-10$. Use FOIL method to expand the product.
$x^2-3x-10$. Use FOIL method to expand the product.
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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. Polynomials are closed under multiplication operations.
The product of two polynomials is always a polynomial. Polynomials are closed under multiplication operations.
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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. Polynomials are closed under multiplication operations.
The product of two polynomials is always a polynomial. Polynomials are closed under multiplication operations.
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What is the definition of a polynomial in $x$?
What is the definition of a polynomial in $x$?
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A finite sum $a_nx^n+\cdots+a_1x+a_0$ with $n$ a whole number. Standard definition combining coefficients, variables, and non-negative integer exponents.
A finite sum $a_nx^n+\cdots+a_1x+a_0$ with $n$ a whole number. Standard definition combining coefficients, variables, and non-negative integer exponents.
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What is a trinomial?
What is a trinomial?
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A polynomial with exactly three terms, such as $x^2+x+1$. Polynomial with exactly three terms joined by operations.
A polynomial with exactly three terms, such as $x^2+x+1$. Polynomial with exactly three terms joined by operations.
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What is a coefficient in the term $-9x^4$?
What is a coefficient in the term $-9x^4$?
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$-9$. The numerical factor multiplying the variable part.
$-9$. The numerical factor multiplying the variable part.
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What is the constant term in the polynomial $3x^2-7x+10$?
What is the constant term in the polynomial $3x^2-7x+10$?
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$10$. The term with no variable, which equals $10$.
$10$. The term with no variable, which equals $10$.
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What is the degree of the monomial $5x^7$?
What is the degree of the monomial $5x^7$?
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$7$. The exponent of the variable in the monomial.
$7$. The exponent of the variable in the monomial.
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What is the degree of the polynomial $4x^3-2x^5+1$?
What is the degree of the polynomial $4x^3-2x^5+1$?
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$5$. The highest exponent among all terms.
$5$. The highest exponent among all terms.
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What is the product $(2x-1)(x+6)$ in standard form?
What is the product $(2x-1)(x+6)$ in standard form?
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$2x^2+11x-6$. Use FOIL method to expand the product.
$2x^2+11x-6$. Use FOIL method to expand the product.
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What is the product $(x-5)(x+2)$ in standard form?
What is the product $(x-5)(x+2)$ in standard form?
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$x^2-3x-10$. Use FOIL method to expand the product.
$x^2-3x-10$. Use FOIL method to expand the product.
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What is the product $(x+4)(x+3)$ in standard form?
What is the product $(x+4)(x+3)$ in standard form?
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$x^2+7x+12$. Use FOIL: First, Outer, Inner, Last terms.
$x^2+7x+12$. Use FOIL: First, Outer, Inner, Last terms.
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What is the product $-2x^2(3x-7)$ in standard form?
What is the product $-2x^2(3x-7)$ in standard form?
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$-6x^3+14x^2$. Distribute $-2x^2$ to each term in the parentheses.
$-6x^3+14x^2$. Distribute $-2x^2$ to each term in the parentheses.
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What is the product $3x(2x^2-5x+4)$ in standard form?
What is the product $3x(2x^2-5x+4)$ in standard form?
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$6x^3-15x^2+12x$. Distribute $3x$ to each term in the parentheses.
$6x^3-15x^2+12x$. Distribute $3x$ to each term in the parentheses.
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What is the result of simplifying $3x^2+5x^2$?
What is the result of simplifying $3x^2+5x^2$?
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$8x^2$. Add coefficients of like terms: $3+5=8$.
$8x^2$. Add coefficients of like terms: $3+5=8$.
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What is the result of simplifying $7x-2x+9$?
What is the result of simplifying $7x-2x+9$?
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$5x+9$. Combine like terms: $7x-2x=5x$.
$5x+9$. Combine like terms: $7x-2x=5x$.
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What is the sum $(2x^2-3x+4)+(x^2+5x-6)$ in standard form?
What is the sum $(2x^2-3x+4)+(x^2+5x-6)$ in standard form?
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$3x^2+2x-2$. Add corresponding terms: $(2+1)x^2+(-3+5)x+(4-6)$.
$3x^2+2x-2$. Add corresponding terms: $(2+1)x^2+(-3+5)x+(4-6)$.
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What is the difference $(5x^2+2x-1)-(3x^2-4x+6)$ in standard form?
What is the difference $(5x^2+2x-1)-(3x^2-4x+6)$ in standard form?
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$2x^2+6x-7$. Distribute negative and combine: $(5-3)x^2+(2+4)x+(-1-6)$.
$2x^2+6x-7$. Distribute negative and combine: $(5-3)x^2+(2+4)x+(-1-6)$.
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What is the product $(x-4)^2$ in standard form?
What is the product $(x-4)^2$ in standard form?
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$x^2-8x+16$. Perfect square trinomial: $(a-b)^2=a^2-2ab+b^2$.
$x^2-8x+16$. Perfect square trinomial: $(a-b)^2=a^2-2ab+b^2$.
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What is the result of simplifying $(x^2-2x)-(3x^2+x)$?
What is the result of simplifying $(x^2-2x)-(3x^2+x)$?
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$-2x^2-3x$. Rewrite as addition then combine like terms.
$-2x^2-3x$. Rewrite as addition then combine like terms.
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What is the result of simplifying $(4x^3-2x+1)+( -x^3+5x-3)$?
What is the result of simplifying $(4x^3-2x+1)+( -x^3+5x-3)$?
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$3x^3+3x-2$. Combine like terms: $(4-1)x^3+(-2+5)x+(1-3)$.
$3x^3+3x-2$. Combine like terms: $(4-1)x^3+(-2+5)x+(1-3)$.
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What is the result of simplifying $(6x^2+1)-(2x^2-3)$?
What is the result of simplifying $(6x^2+1)-(2x^2-3)$?
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$4x^2+4$. Distribute negative and combine: $(6-2)x^2+(1+3)$.
$4x^2+4$. Distribute negative and combine: $(6-2)x^2+(1+3)$.
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What is a binomial?
What is a binomial?
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A polynomial with exactly two terms, such as $x+5$. Polynomial with exactly two terms joined by addition or subtraction.
A polynomial with exactly two terms, such as $x+5$. Polynomial with exactly two terms joined by addition or subtraction.
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What are like terms?
What are like terms?
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Terms with the same variable part, such as $3x^2$ and $-5x^2$. Terms with identical variable parts and exponents.
Terms with the same variable part, such as $3x^2$ and $-5x^2$. Terms with identical variable parts and exponents.
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What is the first step to add or subtract polynomials efficiently?
What is the first step to add or subtract polynomials efficiently?
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Combine like terms. Group and add coefficients of terms with same variable parts.
Combine like terms. Group and add coefficients of terms with same variable parts.
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What property justifies removing parentheses in $-(x^2-3x+2)$?
What property justifies removing parentheses in $-(x^2-3x+2)$?
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Distributive property of multiplication over subtraction. Multiply each term inside by $-1$.
Distributive property of multiplication over subtraction. Multiply each term inside by $-1$.
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What is the additive identity polynomial?
What is the additive identity polynomial?
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$0$. Adding zero to any polynomial leaves it unchanged.
$0$. Adding zero to any polynomial leaves it unchanged.
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What is the multiplicative identity polynomial?
What is the multiplicative identity polynomial?
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$1$. Multiplying by one leaves any polynomial unchanged.
$1$. Multiplying by one leaves any polynomial unchanged.
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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. Polynomials are closed under subtraction operations.
The difference of two polynomials is always a polynomial. Polynomials are closed under subtraction operations.
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What is the additive inverse of the polynomial $P(x)$?
What is the additive inverse of the polynomial $P(x)$?
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$-P(x)$. The polynomial that when added gives zero.
$-P(x)$. The polynomial that when added gives zero.
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