Question

6 trinomial × binomial (2x^2 - 3x + 1)(x - 4)
7 error analysis a student says: (x + 2)(x + 3) = x^2 + 5 explain what the student did wrong and correct the answer.
8 application a rectangle has side lengths ((x + 4)) and ((x + 1)). write a polynomial that represents the area of the rectangle.
checklist: ☐ distributed to every term ☐ added exponents correctly ☐ combined like terms ☐ final answer in standard form

Explanation:

Problem 6: Trinominal × Binomial

Step1: Distribute each term in the trinomial

Multiply \(2x^2\) by \((x - 4)\), \(-3x\) by \((x - 4)\), and \(1\) by \((x - 4)\):
\(2x^2(x - 4) = 2x^3 - 8x^2\)
\(-3x(x - 4) = -3x^2 + 12x\)
\(1(x - 4) = x - 4\)

Step2: Combine like terms

Add the results together:
\(2x^3 - 8x^2 - 3x^2 + 12x + x - 4\)
Combine \(-8x^2\) and \(-3x^2\): \(-11x^2\)
Combine \(12x\) and \(x\): \(13x\)

The student forgot to multiply the middle terms (the "outer" and "inner" terms in the distributive property, or FOIL method) when expanding \((x + 2)(x + 3)\). Instead of only adding the constants, they should have distributed both \(x\) and the constant to each term in the other binomial.

Correct expansion using FOIL:

• First: \(x \cdot x = x^2\)
• Outer: \(x \cdot 3 = 3x\)
• Inner: \(2 \cdot x = 2x\)
• Last: \(2 \cdot 3 = 6\)

Combine like terms: \(x^2 + 3x + 2x + 6 = x^2 + 5x + 6\)

Step1: Recall the area formula for a rectangle

Area \(A = \text{length} \times \text{width}\). Here, length \(= x + 4\) and width \(= x + 1\).

Step2: Expand the product using FOIL

\((x + 4)(x + 1) = x \cdot x + x \cdot 1 + 4 \cdot x + 4 \cdot 1\)
\(= x^2 + x + 4x + 4\)

Step3: Combine like terms

Combine \(x\) and \(4x\): \(5x\)

Answer:

\(2x^3 - 11x^2 + 13x - 4\)

Problem 7: Error Analysis