Problem 134 Factor each polynomial by groupi... [FREE SOLUTION]

Chapter 5: Problem 134

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

Short Answer

Expert verified

(t^2 - 3)(a + 1)

Step by step solution

Arrange the polynomial

Rearrange the polynomial terms in a way that groups can be easily formed. Write the polynomial as: t^{2}a + t^{2} - 3a - 3

Group the terms

Group the terms in pairs to facilitate factoring by grouping:(t^{2}a + t^{2}) - (3a + 3)

Factor out the greatest common factor (GCF) from each group

In the first group (t^{2}a + t^{2}), the GCF is t^2. In the second group (3a + 3), the GCF is 3. Factor the GCF from each group:t^{2}(a + 1) - 3(a + 1)

Factor out the common binomial

Notice the common binomial factor (a + 1) in each group. Factor (a + 1) out of the entire expression:(t^{2} - 3)(a + 1)

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Key Concepts

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

Polynomials

Polynomials are expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication. They do not include division by variables. Each individual part of a polynomial separated by an addition or subtraction sign is known as a term. For instance, in the polynomial $t^2 a - 3 a - 3 + t^2$, the terms are $t^2a$, $-3a$, $-3$, and $t^2$. Polynomials are often categorized by their degree, which is the highest power of the variable in the expression.
Understanding polynomials is crucial for factoring by grouping, as it helps to identify common factors and group terms efficiently.

Greatest Common Factor

The greatest common factor (GCF) of a set of terms is the largest factor that each of the terms share. To factor by grouping, you first need to identify the GCF in each group of terms. For example in the polynomial $t^2a + t^2 - 3a - 3$, we group the terms as $(t^2a + t^2) - (3a + 3)$.
To find the GCF of the first group, $t^2a + t^2$, we see that both terms share a common factor of $t^2$. Hence, we can factor out $t^2$, resulting in $t^2(a + 1)$.
In the second group, $3a + 3$, the common factor is 3, and factoring this out gives us $3(a + 1)$.

Binomial

A binomial is a polynomial that contains exactly two terms. When factoring by grouping, identifying common binomials is essential. In our original polynomial $t^2 a - 3 a - 3 + t^2$, after reformatting and grouping, we get $t^2(a + 1) - 3(a + 1)$.
Here, $a + 1$ is a binomial that is common to both grouped pairs. To complete the factoring by grouping process, we factor out the common binomial factor $a + 1$ from the entire polynomial.
This results in $(t^2 - 3)(a + 1)$. Hence, the original polynomial is factored into two binomials multiplied together.

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91影视

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

Short Answer

Step by step solution

Arrange the polynomial

Group the terms

Factor out the greatest common factor (GCF) from each group

Factor out the common binomial

Key Concepts

Polynomials

Greatest Common Factor

Binomial

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