Question

Use the method of your choice to factor the polynomial completely. Explain your reasoning. $$ z^3-7 z^2-9 z+63 $$

Short answer

The factorized form of the polynomial \(z^3-7 z^2-9 z+63\) is \((z-7)(z+3)(z-3)\).

Step-by-step solution

Identify the Polynomial

The polynomial in this problem is \(z^3-7 z^2-9 z+63\). The aim is to factorize this polynomial.

Factorise

To approach this, we need to check for any common multiples or roots which could be factored out. We can start by grouping and factoring the polynomial which might involve grouping based on similar terms and factoring each group separately. The polynomial given has coefficients that suggest a grouping factorization: \(z^3-7 z^2-9 z+63 = z^2(z-7) -9(z-7)\). After grouping, we see that the factor \(z-7\) is common to both groups and can be factored out, which leads us to our simplified, factored polynomial of \((z-7)(z^2-9)\).

Simplify Further

The factored out polynomial \((z-7)(z^2-9)\) can be simplified further, since \(z^2-9\) is a difference of squares and can be written as \((z+3)(z-3)\). Thus, the final fully factorized form of the polynomial is \((z-7)(z+3)(z-3)\).

Key concepts

Polynomial Factorization

Understanding the technique of polynomial factorization is essential when working with algebraic equations. Factorization involves breaking down a polynomial into a product of simpler polynomials that, when multiplied together, give the original polynomial. To factor a polynomial like \(z^3-7 z^2-9 z+63\), we look for patterns, group terms with common factors, and apply special product rules.

It's helpful to remember certain factorization formulas, like the difference of squares and the perfect square trinomials. Factorization often requires trial and error and a strong grasp of algebraic identities. With practice, you can quickly identify the most efficient factorization method to use.

Difference of Squares

The difference of squares is a specific situation in polynomial factorization defined by the formula \(a^2 - b^2 = (a + b)(a - b)\). This formula tells us that any expression that can be written as one square minus another square can be factored into the product of the sum and the difference of the square roots of those squares.

For example, the expression \(z^2-9\) from the polynomial \(z^3-7 z^2-9 z+63\) is a difference of squares because it can be re-written as \(z^2 - 3^2\), which factors into \((z+3)(z-3)\). Recognizing this pattern is a key step in polynomial factorization.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and arithmetic operations. Polynomial expressions are a type of algebraic expression that can contain constants, coefficients, and one or more variables raised to positive integer powers. The expression \(z^3-7 z^2-9 z+63\) combines variables of different degrees with numerical coefficients.

Manipulating these expressions includes operations such as addition, subtraction, and factoring. The key to mastering algebra is understanding how to transform these expressions into equivalent ones, like factored forms, which often make solving equations easier.

Common Factors

In polynomial factorization, identifying common factors across terms is a critical step. A common factor is a term that is found in each part of the polynomial. In our example, once the polynomial is grouped as \(z^2(z-7) -9(z-7)\), we observe that the binomial \((z-7)\) is a common factor present in both groups.

Factoring out this common binomial simplifies the polynomial into a more manageable format. The goal is to rewrite the original expression as a product of its factors, making the relationships between the terms more transparent and the subsequent algebraic operations simpler to perform.