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Chapter 6: Problem 14

Factor each binomial completely. \(-9 t^{2}+1\)

Short Answer

Expert verified
The factorization of \( -9t^2 + 1 \) is \( -(3t + 1)(3t - 1) \).

Step by step solution

01

Identify the Difference of Squares

The given expression is \( -9t^2 + 1 \). Notice that this is a difference of squares, where \( a^2 - b^2 = (a+b)(a-b) \). We can rewrite the expression as \( -(9t^2 - 1) \) to clearly see the difference of squares.
02

Express As a Difference of Squares

Rewrite \( 9t^2 - 1 \) as \( (3t)^2 - 1^2 \). This confirms it's a difference of squares, where \( a = 3t \) and \( b = 1 \). This means we can factor the expression using the formula for difference of squares.
03

Apply the Difference of Squares Formula

Use the difference of squares formula: \( a^2 - b^2 = (a+b)(a-b) \). Substitute \( a = 3t \) and \( b = 1 \) into the formula to get: \( (3t + 1)(3t - 1) \). Therefore, the expression \( 9t^2 - 1 \) factors to \((3t + 1)(3t - 1) \).
04

Account for the Negative Sign

Remember to include the negative sign initially moved outside when we re-expressed \( -9t^2 + 1 \) as \( -(9t^2 - 1) \). This results in the final factorization: \( -(3t + 1)(3t - 1) \).

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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 key concept in algebra that simplifies certain types of binomial expressions. It is based on the identity \( a^2 - b^2 = (a+b)(a-b) \). This formula indicates that the expression is a product of two conjugate pairs. In simpler terms, it's the difference (subtraction) between two perfect squares.
  • Perfect Square: A number or expression like \( 9t^2 \), which can be expressed as \( (3t)^2 \).
  • Difference: The subtraction sign between two squares, as seen in \( -9t^2 + 1 \) requires us to identify the squares separately.
Recognizing the difference of squares allows us to factor expressions quickly and efficiently, transforming complicated problems into easier ones. Understanding this identity is crucial in the journey of learning algebra.
Algebraic Expressions
Algebraic expressions consist of variables, numbers, and operational symbols forming parts or entire equations. In the expression \(-9t^2 + 1\), the factors of algebraic expressions are critical components that influence the equation's behavior. Let鈥檚 break down what elements form this expression:
  • Terms: The building blocks of the expression, such as \(-9t^2\) and \(+1\).
  • Coefficients: Numbers that multiply the variables, for example, \(-9\) in \(-9t^2\).
  • Variables: Letters representing unknown values like \(t\) in this expression.
  • Constants: Separate numbers in the expression without variables, such as \(+1\).
These components create a structure that can be manipulated using algebraic rules and identities. Proper understanding of such expressions ensures students master the foundation needed for effective problem-solving.
Polynomial Factorization
Polynomial factorization is a method used to simplify polynomials into products of simpler polynomials. In the exercise, the given polynomial \(-9t^2 + 1\) was factored to \(-(3t + 1)(3t - 1)\). Factoring involves:
  • Identifying patterns such as the difference of squares.
  • Rewriting the polynomial in a way that showcases factorable elements.
  • Applying specific formulas or techniques like the difference of squares formula.
When factoring, your goal is to break down a complex expression into pairs of binomials or monomials. This is especially important in algebra and calculus to simplify expression evaluation and solution finding. Lessons in factorization teach students to recognize polynomial patterns and use them skillfully in mathematical equations.

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Most popular questions from this chapter

Factor each trinomial completely. See Examples I through II and Section 6.2. \(1+6 x^{2}+x^{4}\)

Factor each binomial completely. \(125-r^{3}\)

Factor each trinomial completely. $$ z^{2}(x+1)-3 z(x+1)-70(x+1) $$

Factor each trinomial completely. See Examples I through II and Section 6.2. \(3 r^{2}+10 r-8\)

Factor each trinomial completely. See Examples 1 through 11 and Section 6.2. \(-3 t+4 t^{2}-7\)
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