Problem 87 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The trinomial factors to \((x+1)(z-10)(z+7)\).

Step 1: Identify the Common Factor

First, look for any common factors in the expression. Notice that \((x+1)\) is a common factor in all the terms: \(z^2(x+1)\), \(-3z(x+1)\), and \(-70(x+1)\). Thus, we can factor \((x+1)\) out of the entire expression.

Step 2: Factor Out the Common Factor

Factor \((x+1)\) out of the trinomial. This gives: \((x+1)(z^2 - 3z - 70)\). Now, our task is to factor the quadratic expression \(z^2 - 3z - 70\) completely.

Step 3: Factor the Quadratic Expression

Look at the quadratic \(z^2 - 3z - 70\). We need to find two numbers that multiply to \(-70\) and add to \(-3\). These numbers are \(-10\) and \(7\), because \(-10 \times 7 = -70\) and \(-10 + 7 = -3\).

Step 4: Write the Quadratic as a Product of Binomials

Using the numbers \(-10\) and \(7\), write the quadratic \(z^2 - 3z - 70\) as a product of two binomials: \((z - 10)(z + 7)\).

Step 5: Combine All Parts to Get the Fully Factored Expression

Combine the factored quadratic with the factored common term: \((x+1)(z-10)(z+7)\).This is the expression fully factored.

Key Concepts

Common Factor

When working with polynomials, identifying a common factor can simplify the process. A common factor is a value that divides all terms in an expression without leaving a remainder. In the given trinomial, \((x+1)\) is a common factor across each term:

• \(z^2(x+1)\)
• \(-3z(x+1)\)
• \(-70(x+1)\)

By factoring out \((x+1)\), you can reduce the expression to something simpler, making further manipulation easier. This technique streamlines the factoring process and is an important first step when dealing with polynomials.

Quadratic Expression

A quadratic expression is a polynomial of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. The expression we need to factor here is \(z^2 - 3z - 70\). The key part of handling a quadratic is finding two numbers that multiply to give \(ac\) (in this case, \(-70\)) and add up to \(b\) (here, \(-3\)).
This step involves understanding how the terms interact with each other, which is vital for correctly breaking down the expression for further factoring.

Binomial Product

Breaking a quadratic expression into a product of binomials involves writing it as two binomials multiplied together. For \(z^2 - 3z - 70\), we've already found the numbers \(-10\) and \(7\) that satisfy our conditions.
These values help us rewrite the quadratic as a binomial product: \((z - 10)(z + 7)\).
This step checks our understanding of how binomials expand into quadratics and aids in verifying that the expression is correctly factored by ensuring the product of these binomials reproduces the original quadratic.

Factoring Techniques

Factoring techniques like searching for a common factor and employing the quadratic strategy are useful for simplifying expressions and solving equations. The steps in our process involve:

• Recognizing and factoring out shared elements — our common factor.
• Rewriting the quadratic into a binomial product by determining suitable integers \(-10\) and \(7\).
• Combining all parts efficiently for the fully factored form \((x+1)(z-10)(z+7)\).

These techniques are valuable in algebra and can be applied to a wide range of polynomial challenges.