Question

multiplying polynomials

1. to multiply a monomial and a polynomial, multiply the coefficients and add exponents of like variables
2. combine like terms.
3. always write answer in standard form

what is the coefficient of the quadratic term? linear term?

1. $2x^2(3x^2 + 5x - 3)$

b. $(z - 2)(z + 6)$

1. $(2x + 1)(3x - 5)$

c. $(4c - 7)(3c + 8)$
a. $(y + 4)(y + 1)$
d. $(2m - 3)(-6m - 7)$

Explanation:

1. $2x^2(3x^2 + 5x - 3)$

Step1: Distribute monomial to each term

$2x^2 \cdot 3x^2 + 2x^2 \cdot 5x + 2x^2 \cdot (-3)$

Step2: Simplify each product

$6x^4 + 10x^3 - 6x^2$

Step3: Identify coefficients

Quadratic term: $-6x^2$, coefficient = $-6$; Linear term: none, coefficient = $0$

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b. $(z - 2)(z + 6)$

Step1: Apply FOIL method

$z \cdot z + z \cdot 6 + (-2) \cdot z + (-2) \cdot 6$

Step2: Simplify each product

$z^2 + 6z - 2z - 12$

Step3: Combine like terms

$z^2 + 4z - 12$

Step4: Identify coefficients

Quadratic term: $z^2$, coefficient = $1$; Linear term: $4z$, coefficient = $4$

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2. $(2x + 1)(3x - 5)$

Step1: Apply FOIL method

$2x \cdot 3x + 2x \cdot (-5) + 1 \cdot 3x + 1 \cdot (-5)$

Step2: Simplify each product

$6x^2 - 10x + 3x - 5$

Step3: Combine like terms

$6x^2 - 7x - 5$

Step4: Identify coefficients

Quadratic term: $6x^2$, coefficient = $6$; Linear term: $-7x$, coefficient = $-7$

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c. $(4c - 7)(3c + 8)$

Step1: Apply FOIL method

$4c \cdot 3c + 4c \cdot 8 + (-7) \cdot 3c + (-7) \cdot 8$

Step2: Simplify each product

$12c^2 + 32c - 21c - 56$

Step3: Combine like terms

$12c^2 + 11c - 56$

Step4: Identify coefficients

Quadratic term: $12c^2$, coefficient = $12$; Linear term: $11c$, coefficient = $11$

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a. $(y + 4)(y + 1)$

Step1: Apply FOIL method

$y \cdot y + y \cdot 1 + 4 \cdot y + 4 \cdot 1$

Step2: Simplify each product

$y^2 + y + 4y + 4$

Step3: Combine like terms

$y^2 + 5y + 4$

Step4: Identify coefficients

Quadratic term: $y^2$, coefficient = $1$; Linear term: $5y$, coefficient = $5$

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d. $(2m - 3)(-6m - 7)$

Step1: Apply FOIL method

$2m \cdot (-6m) + 2m \cdot (-7) + (-3) \cdot (-6m) + (-3) \cdot (-7)$

Step2: Simplify each product

$-12m^2 - 14m + 18m + 21$

Step3: Combine like terms

$-12m^2 + 4m + 21$

Step4: Identify coefficients

Quadratic term: $-12m^2$, coefficient = $-12$; Linear term: $4m$, coefficient = $4$

Answer:

1. $6x^4 + 10x^3 - 6x^2$; Quadratic coefficient: $-6$, Linear coefficient: $0$

b. $z^2 + 4z - 12$; Quadratic coefficient: $1$, Linear coefficient: $4$

1. $6x^2 - 7x - 5$; Quadratic coefficient: $6$, Linear coefficient: $-7$

c. $12c^2 + 11c - 56$; Quadratic coefficient: $12$, Linear coefficient: $11$
a. $y^2 + 5y + 4$; Quadratic coefficient: $1$, Linear coefficient: $5$
d. $-12m^2 + 4m + 21$; Quadratic coefficient: $-12$, Linear coefficient: $4$