Which Binomial Is A Factor Of 9x2 64

Which binomial isa factor of 9x² – 64? A step‑by‑step guide to factoring a difference of squares

Factoring polynomials is a core skill in algebra, and recognizing patterns such as the difference of squares can save time on exams and in real‑world applications. Think about it: this article explains which binomial is a factor of 9x² – 64, walks you through the reasoning, and provides tools to verify your answer. By the end, you will be able to identify binomial factors of similar expressions confidently.

Introduction

The expression 9x² – 64 appears frequently in algebra worksheets and standardized tests. Understanding its structure allows you to answer the question “which binomial is a factor of 9x² – 64?” quickly. The answer lies in rewriting the polynomial as a difference of squares and then applying the standard factoring formula.

Understanding the Expression

Recognizing the pattern

The terms 9x² and 64 are perfect squares:

• 9x² = (3x)²
• 64 = 8²

When a polynomial can be written as the subtraction of two perfect squares, it is a difference of squares. The generic form is:

[ a^{2} - b^{2} = (a - b)(a + b) ]

Identifying a and b is the first step toward factoring.

Why the pattern matters

• It simplifies the polynomial into a product of two binomials.
• Each binomial becomes a factor that can be used to solve equations or simplify rational expressions.
• The technique is applicable to any expression of the form square – square, regardless of coefficients.

Factoring as a Difference of Squares

Applying the formula

Set a = 3x and b = 8. Substituting into the formula gives:

[ (3x)^{2} - 8^{2} = (3x - 8)(3x + 8) ]

Thus, the original polynomial 9x² – 64 can be rewritten as the product of the two binomials (3x – 8) and (3x + 8).

Which binomial is a factor?

Both (3x – 8) and (3x + 8) are factors, but the question often asks for one binomial factor. That said, either answer is correct if the problem does not specify which one. In many multiple‑choice settings, the option (3x – 8) appears as the correct choice because it matches the standard ordering of the formula (a – b).

How to Verify the Factor

Polynomial long division

To confirm that (3x – 8) divides 9x² – 64 without remainder, perform division:

1. Divide the leading term 9x² by 3x → 3x.
2. Multiply (3x – 8) by 3x → 9x² – 24x.
3. Subtract from the original polynomial → 24x – 64.
4. Divide 24x by 3x → 8.
5. Multiply (3x – 8) by 8 → 24x – 64.
6. Subtract → 0 remainder.

Since the remainder is zero, (3x – 8) is indeed a factor.

For more on this topic, read our article on words before a french kiss crossword or check out words with the root cycle.

Synthetic substitution

Set the binomial equal to zero: 3x – 8 = 0 → x = 8/3.
Plug this value into the original polynomial:

[9\left(\frac{8}{3}\right)^{2} - 64 = 9 \cdot \frac{64}{9} - 64 = 64 - 64 = 0 ]

A zero result confirms that x = 8/3 is a root, meaning (3x – 8) divides the polynomial evenly.

Common Mistakes to Avoid

• Misidentifying the squares: Ensure both terms are perfect squares before applying the formula.
• Sign errors: The minus sign in the original expression must be preserved; switching to a sum of squares would be incorrect.
• Forgetting the coefficient: The coefficient 9 is itself a square (3²), so the square root includes the coefficient 3 attached to x.
• Choosing the wrong binomial in multiple‑choice questions: Both (3x – 8) and (3x + 8) are factors, but test makers often list only one as an option.

Frequently Asked Questions

1. Can the expression be factored further?

No. After rewriting 9x² – 64 as (3x – 8)(3x + 8), each binomial is linear and cannot be factored further over the real numbers.

2. What if the expression were 9x² + 64?

A sum of squares does not factor over the real numbers; it would require complex numbers: (3x + 8i)(3x – 8i).

3. Does the order of the binomials matter?

Mathematically, (3x – 8)(3x + 8) and (3x + 8)(3x – 8) are equivalent because multiplication is commutative. That said, conventional writing places the (a – b) term first.

4. How does this relate to solving equations? Setting each factor to zero gives the roots x = 8/3 and x = –8/3. These solutions are useful in graphing parabolas or solving quadratic equations.

Conclusion

The question “which binomial is a factor of 9x² – 64?” is answered by recognizing the expression as a difference of squares. By rewriting **9x² – 6