Problem 106 Factor. $$ a^{2} p^{2 a}+a^{... [FREE SOLUTION]

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Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 5: Problem 106

Factor. $$ a^{2} p^{2 a}+a^{2} p^{a}-2 a^{2} $$

Short Answer

Expert verified

The factored form is $a^2 (p^a + 2)(p^a - 1)$.

Step by step solution

Factor out the common term

Identify and factor out the greatest common factor from all terms in the expression. In this case, the greatest common factor is $a^2$. This gives:\[a^2 (p^{2a} + p^a - 2)\]

Rewrite the trinomial inside the parentheses

Simplify the trinomial by letting $x = p^a$. Replace all occurrences of $p^a$ in the trinomial:\[a^2 (x^2 + x - 2)\]

Factor the quadratic expression

Factor the quadratic expression $x^2 + x - 2$. We look for two numbers that multiply to $-2$ and add to $1$. These numbers are $2$ and $-1$. Thus, the factorization is:\[x^2 + x - 2 = (x + 2)(x - 1)\]Substitute $x = p^a$ back into the expression:\[a^2 ((p^a + 2)(p^a - 1))\]

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Key Concepts

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

Greatest Common Factor

When starting to factor polynomials, the first step is often identifying and factoring out the Greatest Common Factor (GCF). The GCF is the largest factor that each term in the polynomial shares.
For example, in the polynomial expression we are working with, $a^{2} p^{2a} + a^{2} p^{a} - 2a^{2}$, the common factor for all terms is $a^2$.
Factor out $a^2$: \[a^2 (p^{2a} + p^{a} - 2)\]\
Factoring out the GCF simplifies the equation and prepares it for further factorization.
This step is crucial for breaking down more complex polynomials.

Quadratic Expressions

A quadratic expression is a polynomial where the highest exponent of the variable is 2. These expressions follow the general format $ax^2 + bx + c$.
In our case, inside the parentheses, the expression is $p^{2a} + p^{a} - 2$. We can simplify handling it by substituting $x = p^a$, turning it into $x^2 + x - 2$.
This simple notation helps you see the quadratic nature of the expression clearly.
The goal is to factor this quadratic expression into simpler binomials.

Substitution Method

The substitution method is a powerful technique for simplifying complex expressions. In this method, you replace a more complicated part of the expression with a simpler variable.
For instance, in our polynomial, instead of working directly with $p^{a}$, we substitute it with $x$. This makes dealing with the quadratic expression easier: \[a^2 (x^2 + x - 2)\]\
Once the expression is factored, we simply substitute back the original values into the solution.
This makes the problem-solving process more manageable and error-free.

Trinomial Factorization

Trinomial factorization is the process of breaking down a quadratic trinomial into two binomials. For example, we need to factor $x^2 + x - 2$.
To do this, find two numbers that multiply together to give the constant term (-2) and add to give the middle coefficient (1). In this case, the numbers are 2 and -1.
Rewrite the quadratic as: \[x^2 + x - 2 = (x + 2)(x - 1)\]\
Finally, substitute $x = p^a$ back into the factored form: \[a^2 ((p^a + 2)(p^a - 1))\]\
This completes the factorization and provides a simplified view of the polynomial.

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91影视

Factor. $$ a^{2} p^{2 a}+a^{2} p^{a}-2 a^{2} $$

Short Answer

Step by step solution

Factor out the common term

Rewrite the trinomial inside the parentheses

Factor the quadratic expression

Key Concepts

Greatest Common Factor

Quadratic Expressions

Substitution Method

Trinomial Factorization

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