14-6 Multiplying a Polynomial by a Monomial

14-6: Multiplying a polynomial by a monomial

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Let's first review things we need to know in order to be successful in multiplying a polynomial by a monomial.

TERMS

monomial: an expression with one term (monocle, monopoly, monorail)
binomial: an expression with two terms (biannual, bicycle, bilingual)
trinomial: an expression with three terms (tricycle, triangle, tripod)
polynomial: any expression with multiple terms (polyester, polygon)

Multiple Select

Select all the monomials.

• $$3x^2+2x-7$$  

• $$10x+2$$  

• $$-7a^2c^4$$  

• $$5x+4y+9xy-10$$  

• $$6x$$  

Multiple Select

Select all the polynomials.

• $$3x^2-7x+4$$

• $$\frac{1}{4}x^4-\frac{2}{3}x^2+\frac{1}{2}x-10$$

• $$6x$$

• $$-13a^2b^4$$

• $$10x+5y$$

Multiple Choice

Simplify: $$3n^5\cdot5n^{10}$$  

• $$15n^5$$  

• $$8n^{15}$$  

• $$15n^{50}$$  

• $$15n^{15}$$  

Multiple Choice

$$Simplify:\ 10xy^3\cdot8x^5y^3$$  

• 80xy³

• 80x⁶y³

• 80x⁶y⁶

• none of the above

Multiple Choice

Simplify -3(2c - 5)

• -6c - 15

• -6c + 15

• 6c + 15

• 6c - 15

How to Multiply a Polynomial by a Monomial

• First, distribute the monomial to each term in the polynomial

• Then, multiply!

Another example...

First, distribute. Then, multiply!

Multiple Choice

Use Distributive Property to solve.

x³(2x²+3x)

• 2x⁵+3x⁴

• 2x⁶+3x³

• 5x⁹

• 5x

Multiple Choice

Multiply 4m(3m⁷ + 2m⁵)

• 7m⁷ + 6m⁶

• 12m⁸ + 8m⁶

Multiple Choice

Simplify:

3a( 5a² + 8a + 2 )

• 15a³ + 24a² + 6a

• 10a³ + 24a + 6

• 8a³ + 11a² + 6a

• 15a² + 24a + 6

Multiple Choice

-5x⁶(3x² - 12x + 30)

• -15x⁸ + 60x⁷ - 150x⁶

• 15x⁸ - 60x⁷ - 150x⁶

• 8x⁶ - 17x + 35

• -8x⁶ + 17x - 35x²

Multiple Choice

Multiply 7x²y³ (2x⁴y⁵+6xy³)

• 14x⁸y¹⁵+42x²y⁹

• 9x⁶y⁸+13x³y⁶

• 14x⁶y⁸+42x³y⁶

• 14x⁶y⁸+42x²y⁶

Simplifying Expressions with Polynomials

Sometimes, there are TWO or more instances where we need to use the distributive property to simplify an expression.

Remember when simplifying any expression...

• First, use the Distributive Property to get rid of the parentheses.

• Then, combine like terms. (Same variable to the same exponent)

• An expression is fully simplified when it has no like terms and no exponents.

Multiple Choice

Simplify the expression

$$-4d\left(5d^2-12\right)+7\left(d+5\right)$$  

• $$-20d^3+55d+35$$  

• $$-20d^3+48d+7d+35$$  

• $$-20d^3-41d+35$$  

Multiple Choice

Simplify the expression

$$-3\left(5x^2+2x+9\right)+x\left(2x-3\right)$$  

• $$13x^2+3x-27$$  

• $$-17x^2+3x+27$$  

• $$-13x^2-9x-27$$  

Multiple Choice

Solve the equation for y

$$3\left(y-2\right)+2y=4y+14$$  

• $$y=\frac{20}{9}$$  

• $$y=20$$  

• $$y=8$$  

Multiple Choice

Any expression with multiple terms can be referred to as what?

• polynomial

• monomial

• manynomial

• trinomial