Question

multiplying polynomials
independent practice
name
date
pd
use your knowledge of multiplying polynomials to solve the problems below.

1. three students used different methods to multiply polynomials. carefully observe each student’s work and describe the mistake that was made.

hector
$(x - 6)(x - 9)$
table with x, 6 and x, 9 as rows and columns, with cells $x^2$, $6x$, $9x$, 54 and expression $x^2 + 15x + 54$
hector’s mistake:

jenny
$(3x + 5)(4x - 5)$
$3x(4x - 5)+5(4x - 5)$
$7x^2 - 2x + 9x - 25$
$7x^2 + 7x - 25$
jenny’s mistake:

elias
$(x^2 - 3)(x + 2)$
diagram of multiplying and result $x^3 - 6$
elias’ mistake:

use any method to find the product of each of the following
2.
$(x + 7)(x - 3)$
3.
$(-6x - 1)(4x - 3)$
4.
$(x^2 + 8)(-3x + 5)$

Explanation:

1. Analyzing Each Student's Mistake

Hector's Work:

• Problem: Multiply \((x - 6)(x - 9)\)
• Hector’s Method: Used a table (area model) with columns \(x, 6\) and rows \(x, 9\).
• Mistake: The original binomial is \((x - 6)(x - 9)\), so the terms should be \(x, -6\) (not \(+6\)) and \(x, -9\) (not \(+9\)). By using \(+6\) and \(+9\), he effectively multiplied \((x + 6)(x + 9)\) (which gives \(x^2 + 15x + 54\)), but the problem requires \((x - 6)(x - 9)\).

Jenny's Work:

• Problem: Multiply \((3x + 5)(4x - 5)\)
• Jenny’s Method: Distributed \(3x\) and \(5\) over \((4x - 5)\): \(3x(4x - 5) + 5(4x - 5)\).
• Mistake: When expanding \(3x(4x - 5)\), the product of \(3x\) and \(4x\) is \(12x^2\) (not \(7x^2\)), and \(3x(-5) = -15x\) (not \(-2x\)). She made arithmetic errors in the first distribution step.

Elias's Work:

• Problem: Multiply \((x^2 - 3)(x + 2)\)
• Elias’s Method: Used a visual (likely FOIL or distribution) but simplified incorrectly.
• Mistake: To multiply \((x^2 - 3)(x + 2)\), distribute \(x^2\) and \(-3\) over \((x + 2)\):

\(x^2(x) + x^2(2) - 3(x) - 3(2) = x^3 + 2x^2 - 3x - 6\).
Elias only multiplied \(x^2 \cdot x = x^3\) and \(-3 \cdot 2 = -6\), ignoring the middle terms (\(2x^2\) and \(-3x\)).

2. Finding the Product of Polynomials

Problem 2: \((x + 7)(x - 3)\)

Step1: Use the distributive property (FOIL)

Multiply the First, Outer, Inner, and Last terms:
\(x \cdot x + x \cdot (-3) + 7 \cdot x + 7 \cdot (-3)\)

Step2: Simplify each term

\(x^2 - 3x + 7x - 21\)

Step3: Combine like terms

\(x^2 + 4x - 21\)

Step1: Distribute each term

\(-6x(4x) + (-6x)(-3) + (-1)(4x) + (-1)(-3)\)

Step2: Simplify each term

\(-24x^2 + 18x - 4x + 3\)

Step3: Combine like terms

\(-24x^2 + 14x + 3\)

Step1: Distribute each term

\(x^2(-3x) + x^2(5) + 8(-3x) + 8(5)\)

Step2: Simplify each term

\(-3x^3 + 5x^2 - 24x + 40\)

Step3: No like terms to combine (already simplified)

Answer:

\(x^2 + 4x - 21\)

Problem 3: \((-6x - 1)(4x - 3)\)