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Chapter 5: Problem 64

Factor If \(P(x)=x^{3},\) use factoring to simplify $$ P(a+h)-P(a) $$

Short Answer

Expert verified
The factored form is P(a+h) - P(a) = h(3a^2 + 3ah + h^2).

Step by step solution

01

Substitute Values

First, start by substituting the expressions for the polynomial given. Substitute into the polynomial function to get: P(a+h) = (a + h)^3 P(a) = a^3
02

Compute P(a+h)

Expand the expression (a + h)^3 using the binomial theorem: (a + h)^3 = a^3 + 3a^2h + 3ah^2 + h^3
03

Subtract P(a)

Subtract P(a) from P(a + h): P(a+h) - P(a) = (a^3 + 3a^2h + 3ah^2 + h^3) - a^3
04

Simplify the Expression

Combine like terms to simplify the expression: P(a+h) - P(a) = a^3 + 3a^2h + 3ah^2 + h^3 - a^3 = 3a^2h + 3ah^2 + h^3
05

Factor out h

Observe that the simplified expression has a common factor of h. Factor out h from every term: 3a^2h + 3ah^2 + h^3 = h(3a^2 + 3ah + h^3)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Expansion
Binomial Expansion is a method used to expand expressions raised to a power. It is particularly handy in algebra when dealing with polynomials like \((a + h)^3\). By using the binomial theorem, you can expand such expressions quickly and accurately. The binomial theorem states that for any positive integer n:
\((x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k\)
Here鈥檚 an easy step-by-step breakdown:
  • Identify the terms: These are \(a\) and \(h\) in our case.
  • Identify the power: It鈥檚 3 in this exercise.
  • Use the binomial coefficients: For \((a+h)^3\), expand it to \(a^3 + 3a^2h + 3ah^2 + h^3\).
This approach helps us simplify higher power algebraic expressions. In this problem, expanding \((a+h)^3\) allowed us to see the individual terms clearly and set up for polynomial subtraction.
Polynomial Subtraction
Polynomial Subtraction involves finding the difference between two polynomials. In our exercise, we need to subtract \(P(a) = a^3\) from \(P(a+h) = a^3 + 3a^2h + 3ah^2 + h^3\). Here鈥檚 how you can do it step-by-step:
  • Write down both polynomials: \(P(a + h)\) and \(P(a)\).
  • Align like terms: Place each term next to its like term.
  • Subtract the polynomials: \(P(a + h) - P(a)\). This simplifies to \(a^3 + 3a^2h + 3ah^2 + h^3 - a^3\). Notice how \(a^3\) cancels out.
After subtraction, we are left with \(3a^2h + 3ah^2 + h^3\). Subtracting polynomials is a key skill to have in algebra, as it allows us to simplify expressions and solve equations more easily.
Common Factor
A Common Factor is a factor shared by all terms in an expression. In our simplified expression \(3a^2h + 3ah^2 + h^3\), we can see that \(h\) is a common factor in all three terms. Here鈥檚 how to factor it out:
  • Identify the common factor in each term: In this case, it鈥檚 \(h\).
  • Divide each term by the common factor: \(\frac{3a^2h}{h} + \frac{3ah^2}{h} + \frac{h^3}{h}\).
  • Factor it out: We get \(h(3a^2 + 3ah + h^2)\).
Factoring out the common factor is essential in simplifying algebraic expressions. It makes complex expressions more manageable and can help in solving equations. Factoring is a crucial step in many algebraic processes, offering a clearer, simplified form of our expressions.

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Most popular questions from this chapter

A Pythagorean triple is a set of three numbers that satisfy the Pythagorean equation. They can be generated by choosing natural numbers \(n\) and \(m\) \(n>m,\) and forming the following three numbers: \(n^{2}+m^{2}, n^{2}-m^{2},\) and \(2 m n .\) Show that these three expressions satisfy the Pythagorean equation.

Factor completely. $$ -9 x^{3}+12 x $$

Factor completely. $$ (m-1)^{3}-(m+1)^{3} $$

Solve. If each of the sides of a square is lengthened by 4 \(\mathrm{m}\), the area becomes \(49 \mathrm{m}^{2} .\) Find the length of a side of the original square.

Family has factored a polynomial as \((a-b)(x-y)\) while Jorge has factored the same polynomial as \((b-a)(y-x) .\) Can they both be correct? Why or why not?
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