Exercise 5.3 — Question 1

Topic Overview

This question applies polynomial division, the Remainder Theorem, and the Factor Theorem to find all dimensions of a rectangular solid whose volume is given as a cubic polynomial.

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Key Concepts

Polynomial Division (SLO M-11-A-42)

To divide a polynomial $P(x)$ by a linear factor $(x-k)$, use synthetic division or long division. The result is:

$P(x)=(x-k)\cdot Q(x)+R$

where $Q(x)$ is the quotient and $R$ is the remainder.

Remainder Theorem (SLO M-11-A-43)

Example: For $P(x)=x^3-4x^2+3x-2$ divided by $(x-3)$: $P(3)=27-36+9-2=-2$ So the remainder is $-2$.

Factor Theorem (SLO M-11-A-44)

This is a special case of the Remainder Theorem where the remainder equals zero.

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Worked Example — Q1

Problem: The volume of a rectangular solid is $V(y)=y^3-2y^2-y+2$ One dimension (side length) is $(y-2)$. Find the other two dimensions.

Step 1 — Verify $(y-2)$ is a factor

By the Factor Theorem, check $V(2)$: $V(2)=(2{)}^3-2(2{)}^2-(2)+2=8-8-2+2=0$ Since $V(2)=0$, $(y-2)$ is indeed a factor. ✓

Step 2 — Divide $V(y)$ by $(y-2)$

Using synthetic division with $k=2$ and coefficients $1,-2,-1,2$:

Quotient: $Q(y)=y^2+0y-1=y^2-1$, Remainder $=0$.

So: $V(y)=(y-2)(y^2-1)$

Step 3 — Factor the quadratic

$y^2-1=(y-1)(y+1)$

Result

$V(y)=(y-2)(y-1)(y+1)$

The three dimensions of the rectangular solid are: $\boxed{(y+1),(y-1),(y-2)}$

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Steps to Factorize a Cubic Polynomial Using the Factor Theorem

1. Find a root $k$ such that $P(k)=0$ by testing integer factors of the constant term.
2. Divide $P(x)$ by $(x-k)$ using synthetic or long division to get a quadratic $Q(x)$.
3. Factor $Q(x)$ by inspection, completing the square, or the quadratic formula.
4. Write the full factorization: $P(x)=(x-k)\cdot Q(x)$.