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Chapter 5: Problem 167

Factor each polynomial completely. $$a^{2} x^{2}-b^{2}$$

Short Answer

Expert verified
The factored form is \((ax + b)(ax - b)\).

Step by step solution

01

Identify the Polynomial Structure

Recognize that the given polynomial \(a^2 x^2 - b^2\) is a difference of squares.
02

Recall the Difference of Squares Formula

Recall the difference of squares formula: \(a^2 - b^2 = (a + b)(a - b)\).
03

Rewrite the Polynomial in the Formula

Express the given polynomial in a form that matches the difference of squares formula. Here, \(a^2 x^2\) can be written as \((ax)^2\). So, the polynomial is \((ax)^2 - b^2\).
04

Apply the Difference of Squares Formula

Using the difference of squares formula, factor \((ax)^2 - b^2\) into \((ax + b)(ax - b)\).

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Key Concepts

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

Difference of Squares
The 'difference of squares' is a powerful technique in factoring polynomials. This method applies to expressions in the form of \(a^2 - b^2\).
The term 'difference' indicates subtraction between two square terms.
For instance, in the polynomial \(a^2 x^2 - b^2\), you have two square terms separated by a minus sign.
Remember the difference of squares formula: \(a^2 - b^2 = (a + b)(a - b)\). This formula is fundamental and very useful in algebra.
By recognizing that a polynomial fits this pattern, you can apply the formula to factor the expression efficiently.
Always ensure both terms are perfect squares to use this technique.
Polynomial Structure
Understanding the structure of a polynomial is crucial for factoring. In the polynomial \(a^2 x^2 - b^2\), note the following:
  • \(a^2 x^2\) is a product of \(a^2\) and \(x^2\), both are perfect squares.
  • \(b^2\) is another perfect square.
Knowing the form helps you recognize patterns.
In this case, rewrite the polynomial to fit the difference of squares model. Here, \(a^2 x^2\) can be seen as \((ax)^2\), fitting into \((a x)^2 - b^2\). By expressing the polynomial this way, you align it perfectly with the difference of squares formula.
This step simplifies the problem and makes factoring straightforward.
Factoring Techniques
Factoring techniques like the difference of squares simplify complex polynomials.
Here鈥檚 how to factor the polynomial given in the exercise:
First, recognize that \(a^2 x^2\) can be rewritten as \((ax)^2\). Now, you have \((ax)^2 - b^2\).
Next, apply the difference of squares formula, \(a^2 - b^2 = (a + b)(a - b)\).
These steps transform the polynomial into \( (ax + b)(ax - b) \).
By practicing and mastering this technique, you鈥檒l be able to factor similar polynomials easily.
  • Identify perfect squares.
  • Rewrite the polynomial to match the formula.
  • Apply the difference of squares formula.
This systematic approach ensures accuracy and efficiency.

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Most popular questions from this chapter

Tossing a ball. A boy tosses a ball upward at 32 feet per second from a window that is 48 feet above the ground. The height of the ball above the ground (in feet) at time \(t\) (in seconds) is given by $$h(t)=-16 t^{2}+32 t+48$$ Find the time at which the ball strikes the ground.

New dimensions in gardening. Mary Gold has a rectangular flower bed that measures 4 feet by 6 feet. If she wants to increase the length and width by the same amount to have a flower bed of 48 square feet, then what will be the new dimensions? (GRAPH CANNOT COPY).

Factor each polynomial using the trial-and-error method. $$ b^{2}+2 b-48 $$

Factor each polynomial using the trial-and-error method. $$ 5 y^{2}-14 y-3 $$

Factor each polynomial by grouping. $$ 2 m^{2}-3 a m-3 a t+2 m t $$
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