Question

essential question
how do i multiply monomials and polynomials?
monomial × polynomial

• to multiply a polynomial by a monomial, simply distribute the monomial to each term in the polynomial.
• for terms with exponents, use the product rule → add exponents

product rule: $x^a \cdot x^b = x^{a+b}$
$(3x^2y)(12xy)$
$36x^3y^2$

1. $5m(m^2 - 10m - 4)$
2. $-4x^2(2x - 7 + 3x^2)$
3. $-3x^4(6x^2y^2 - 12xy + 16)$
4. $c^2d(-2c^4 - cd + 3d^3)$
5. $5x(-7x + 3) - \frac{3}{4}x(12 - 8x^2)$
6. $-2x(x^3 - 6x^2 + 6) + 4x^3 - (5x^2 + 10x)$
7. write the area of the trapezoid as a simplified expression.
8. write the area of the shaded region as a simplified expression.

Explanation:

Step1: Distribute monomial to each term

$5m \cdot m^2 - 5m \cdot 10m - 5m \cdot 4$

Step2: Simplify using exponent rule

$5m^3 - 50m^2 - 20m$

Step1: Distribute monomial to each term

$-4x^2 \cdot 2x + (-4x^2) \cdot (-7) + (-4x^2) \cdot 3x^2$

Step2: Simplify using exponent rule

$-8x^3 + 28x^2 - 12x^4$

Step3: Rearrange terms by degree

$-12x^4 - 8x^3 + 28x^2$

Step1: Distribute monomial to each term

$-3x^4 \cdot 6x^2y^2 + (-3x^4) \cdot (-12xy) + (-3x^4) \cdot 16$

Step2: Simplify using exponent rule

$-18x^6y^2 + 36x^5y - 48x^4$

Step1: Distribute monomial to each term

$c^2d \cdot (-2c^4) + c^2d \cdot (-cd) + c^2d \cdot 3d^3$

Step2: Simplify using exponent rule

$-2c^6d - c^3d^2 + 3c^2d^4$

Step1: Distribute monomials to each term

$5x \cdot (-7x) + 5x \cdot 3 - \frac{3}{4}x \cdot 12 + \frac{3}{4}x \cdot 8x^2$

Step2: Simplify each product

$-35x^2 + 15x - 9x + 6x^3$

Step3: Combine like terms

$6x^3 - 35x^2 + 6x$

Step1: Distribute monomials to each term

$-2x \cdot x^3 + (-2x) \cdot (-6x^2) + (-2x) \cdot 6 + 4x^3 - 5x^2 - 10x$

Step2: Simplify each product

$-2x^4 + 12x^3 - 12x + 4x^3 - 5x^2 - 10x$

Step3: Combine like terms

$-2x^4 + 16x^3 - 5x^2 - 22x$

Step1: Recall trapezoid area formula

$A = \frac{1}{2}(b_1 + b_2)h$

Step2: Substitute given values

$A = \frac{1}{2}[(8x^2 - 12) + (4x^2 + 2x)] \cdot 3x$

Step3: Combine like terms in brackets

$A = \frac{1}{2}(12x^2 + 2x - 12) \cdot 3x$

Step4: Distribute the $\frac{1}{2}$

$A = (6x^2 + x - 6) \cdot 3x$

Step5: Distribute the $3x$

$A = 18x^3 + 3x^2 - 18x$

Step1: Area of outer rectangle minus inner

$A = (9x+4)(5x) - (3x-5)(2x)$

Step2: Simplify each product

$A = 45x^2 + 20x - (6x^2 - 10x)$

Step3: Distribute the negative sign

$A = 45x^2 + 20x - 6x^2 + 10x$

Step4: Combine like terms

$A = 39x^2 + 30x$

Answer:

1. $5m^3 - 50m^2 - 20m$
2. $-12x^4 - 8x^3 + 28x^2$
3. $-18x^6y^2 + 36x^5y - 48x^4$
4. $-2c^6d - c^3d^2 + 3c^2d^4$
5. $6x^3 - 35x^2 + 6x$
6. $-2x^4 + 16x^3 - 5x^2 - 22x$
7. $18x^3 + 3x^2 - 18x$
8. $39x^2 + 30x$