Master Multiplying Polynomials with Distributive Property and FOIL Method

Polynomial	Degree	Leading Coefficient
$P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}$	$n$	$a_{n}$
$C \cdot P (x) = C \cdot a_{n} x^{n} + C \cdot a_{n - 1} x^{n - 1} + \dots + C \cdot a_{1} x + C \cdot a_{0}$	$n$	$C \cdot a_{n}$

Polynomial

Degree

Leading Coefficient

P (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}

n

a_{n}

C \cdot P (x) = C \cdot a_{n} x^{n} + C \cdot a_{n - 1} x^{n - 1} + \dots + C \cdot a_{1} x + C \cdot a_{0}

n

C \cdot a_{n}

a The volume of a rectangular container is found by multiplying its three dimensions.

V = w \cdot ℓ \cdot h

Therefore, to find a polynomial representing the container's volume after passing

t

hours underwater, multiply the expressions that model the dimensions of the container.

w (t) \cdot ℓ (t) \cdot h (t) ⇓ (\frac{1}{4} t^{2} + 5) (- \frac{1}{9} t^{2} + 2 t + 2) (t + 3)

By applying the Commutative Property of Multiplication, the previous product can be rewritten to have the two binomials next to each other.

(- \frac{1}{9} t^{2} + 2 t + 2) (\frac{1}{4} t^{2} + 5) (t + 3)

Then, these two binomials can be multiplied using the FOIL method. For simplicity, the trinomial will not be written during this multiplication.

Start by multiplying the first terms.
$(\frac{1}{4} t^{2} + 5) (t + 3) = \frac{1}{4} t^{2} \cdot t + \dots$
Continue by multiplying the outer terms.
$(\frac{1}{4} t^{2} + 5) (t + 3) = \frac{1}{4} t^{2} \cdot t + \frac{1}{4} t^{2} \cdot 3 + \dots$
Next, multiply the inner terms.
$(\frac{1}{4} t^{2} + 5) (t + 3) = \frac{1}{4} t^{2} \cdot t + \frac{1}{4} t^{2} \cdot 3 + 5 \cdot t + \dots$
Now, multiply the last terms.
$(\frac{1}{4} t^{2} + 5) (t + 3) = \frac{1}{4} t^{2} \cdot t + \frac{1}{4} t^{2} \cdot 3 + 5 \cdot t + 5 \cdot 3$
Finally, simplify the resulting expression by performing the required multiplications.
$(\frac{1}{4} t^{2} + 5) (t + 3) = \frac{1}{4} t^{3} + \frac{3}{4} t^{2} + 5 t + 15$

Finally, the trinomial has to be multiplied by the resulting polynomial. To perform such multiplication the Distributive Property can be used. Here, each term of the trinomial will be distributed to each term of the second polynomial.

V (t) = (- \frac{1}{9} t^{2} + 2 t + 2) (\frac{1}{4} t^{3} + \frac{3}{4} t^{2} + 5 t + 15)

Multiply

(- \frac{1}{9} t^{2} + 2 t + 2)

(\frac{1}{4} t^{3} + \frac{3}{4} t^{2} + 5 t + 15)

Distr

Distribute $- \frac{1}{9} t^{2} + 2 t + 2$

V (t) = (- \frac{1}{9} t^{2} + 2 t + 2) \frac{1}{4} t^{3} + (- \frac{1}{9} t^{2} + 2 t + 2) \frac{3}{4} t^{2} + (- \frac{1}{9} t^{2} + 2 t + 2) 5 t + (- \frac{1}{9} t^{2} + 2 t + 2) 15

Distr

Distribute $\frac{1}{4} t^{3}, \frac{3}{4} t^{2}, 5 t, & 15$

V (t) = (- \frac{1}{9} t^{2} \cdot \frac{1}{4} t^{3} + 2 t \cdot \frac{1}{4} t^{3} + 2 \cdot \frac{1}{4} t^{3}) + (- \frac{1}{9} t^{2} \cdot \frac{3}{4} t^{2} + 2 t \cdot \frac{3}{4} t^{2} + 2 \cdot \frac{3}{4} t^{2}) + (- \frac{1}{9} t^{2} \cdot 5 t + 2 t \cdot 5 t + 2 \cdot 5 t) + (- \frac{1}{9} t^{2} \cdot 15 + 2 t \cdot 15 + 2 \cdot 15)

CommutativePropMult

\CommutativePropMult

V (t) = (- \frac{1}{9} \cdot \frac{1}{4} t^{2} \cdot t^{3} + 2 \cdot \frac{1}{4} t \cdot t^{3} + 2 \cdot \frac{1}{4} t^{3}) + (- \frac{1}{9} \cdot \frac{3}{4} t^{2} \cdot t^{2} + 2 \cdot \frac{3}{4} t \cdot t^{2} + 2 \cdot \frac{3}{4} t^{2}) + (- \frac{1}{9} \cdot 5 t^{2} \cdot t + 2 \cdot 5 t \cdot t + 2 \cdot 5 t) + (- \frac{1}{9} \cdot 15 t^{2} + 2 \cdot 15 t + 2 \cdot 15)

Simplify right-hand side

MultPow

$a^{m} \cdot a^{n} = a^{m + n}$

V (t) = (- \frac{1}{9} \cdot \frac{1}{4} t^{5} + 2 \cdot \frac{1}{4} t^{4} + 2 \cdot \frac{1}{4} t^{3}) + (- \frac{1}{9} \cdot \frac{3}{4} t^{4} + 2 \cdot \frac{3}{4} t^{3} + 2 \cdot \frac{3}{4} t^{2}) + (- \frac{1}{9} \cdot 5 t^{3} + 2 \cdot 5 t^{2} + 2 \cdot 5 t) + (- \frac{1}{9} \cdot 15 t^{2} + 2 \cdot 15 t + 2 \cdot 15)

MultFrac

Multiply fractions

V (t) = (- \frac{1}{3 6} t^{5} + \frac{2}{4} t^{4} + \frac{2}{4} t^{3}) + (- \frac{3}{3 6} t^{4} + \frac{6}{4} t^{3} + \frac{6}{4} t^{2}) + (- \frac{5}{9} t^{3} + 2 \cdot 5 t^{2} + 2 \cdot 5 t) + (- \frac{1 5}{9} t^{2} + 2 \cdot 15 t + 2 \cdot 15)

SimpQuotProd

Simplify quotient and product

V (t) = (- \frac{1}{3 6} t^{5} + \frac{1}{2} t^{4} + \frac{1}{2} t^{3}) + (- \frac{1}{1 2} t^{4} + \frac{3}{2} t^{3} + \frac{3}{2} t^{2}) + (- \frac{5}{9} t^{3} + 10 t^{2} + 10 t) + (- \frac{5}{3} t^{2} + 30 t + 30)

CommutativePropAdd

\CommutativePropAdd

V (t) = - \frac{1}{3 6} t^{5} + (\frac{1}{2} t^{4} - \frac{1}{1 2} t^{4}) + (\frac{1}{2} t^{3} - \frac{5}{9} t^{3} + \frac{3}{2} t^{3}) + (\frac{3}{2} t^{2} - \frac{5}{3} t^{2} + 10 t^{2}) + (10 t + 30 t) + 30

AddSubTerms

Add and subtract terms

V (t) = - \frac{1}{3 6} t^{5} + \frac{5}{1 2} t^{4} + \frac{1 3}{9} t^{3} + \frac{5 9}{6} t^{2} + 40 t + 30

b The polynomial

V (t)

found in Part A gives the volume of the container after passing

t

hours underwater. It is given that Izabella's nephew put the container underwater at

10 : 00 AM

and took it out at

4 : 00 PM .

Therefore, the container was underwater for

6

hours.

6 hours 10 : 00 AM ⟶ 4 : 00 PM

Consequently, to find the current volume of the container, evaluate

V (t)

t = 6 .

V (t) = - \frac{1}{3 6} t^{5} + \frac{5}{1 2} t^{4} + \frac{1 3}{9} t^{3} + \frac{5 9}{6} t^{2} + 40 t + 30

Substitute

6

for

t

and evaluate

Substitute

$t = 6$

V (6) = - \frac{1}{3 6} (6)^{5} + \frac{5}{1 2} (6)^{4} + \frac{1 3}{9} (6)^{3} + \frac{5 9}{6} (6)^{2} + 40 (6) + 30

CalcPow

Calculate power

V (6) = - \frac{1}{3 6} \cdot 7776 + \frac{5}{1 2} \cdot 1296 + \frac{1 3}{9} \cdot 216 + \frac{5 9}{6} \cdot 36 + 40 (6) + 30

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

V (6) = - \frac{7 7 7 6}{3 6} + \frac{5 \cdot 1 2 9 6}{1 2} + \frac{1 3 \cdot 2 1 6}{9} + \frac{5 9 \cdot 3 6}{6} + 40 (6) + 30

Multiply

V (6) = - \frac{7 7 7 6}{3 6} + \frac{6 4 8 0}{1 2} + \frac{2 8 0 8}{9} + \frac{2 1 2 4}{6} + 240 + 30

CalcQuot

Calculate quotient

V (6) = - 216 + 540 + 312 + 354 + 240 + 30

AddSubTerms

Add and subtract terms

V (6) = 1260

After passing

6

hours underwater, the container will have a volume of

1260

cubic centimeters.

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Catch-Up and Review

Challenge

Investigating the Product of Polynomials

Discussion

Multiplying a Polynomial by a Constant

Example

Finding the Area of a Poster

Hint

Solution

Discussion

Methods for Multiplying Polynomials

Example

Modeling the Pigpen Area

Hint

Solution

Example

Determining a Toy's Volume

Hint

Solution

Discussion

Analyzing the Product of Two Polynomials

Rule

Closure Property of Polynomial Multiplication

Proof

Pop Quiz

Practicing Multiplication of Polynomials

Closure

Computing a Polynomials Product

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