Problem 13 - Precalculus : Building Concepts and Connections

Question

Show that the given value of \(x\) is a zero of the polynomial. Use the zero to completely factor the polynomial. $$p(x)=-x^{4}-x^{3}+18 x^{2}+16 x-32 ; x=1$$

Step-by-Step Solution

Answer

The polynomial \( p(x) = -x^{4}-x^{3}+18x^{2}+16x-32 \) factors as \(-1(x - 1)(x^{3} -2x^{2} +20x +32)\) if \(x = 1\) is indeed a zero of it.

Step 1: Verify the Given Zero

Substitute \(x = 1\) in \(p(x) = -x^{4}-x^{3}+18x^{2}+16x-32\). If \(p(1) = 0\), then \(x = 1\) is indeed a zero of the polynomial.

Step 2: Perform Synthetic Division

Use synthetic division to divide the polynomial by \(x - 1\). Write the coefficients of the polynomial at the top of the synthetic division bracket (-1, -1, 18, 16, -32) and the zero (1) to the left of the bracket. Carry out the synthetic division process to find the other factor of the polynomial.

Step 3: Factor the Polynomial Remaining after Synthetic Division

The result from synthetic division represents the coefficients of the remaining polynomial. Write down this polynomial and factor it using methods of factoring quadratic expressions. If the remaining polynomial can't be factored further, it is a prime polynomial.

Key Concepts

Understanding Synthetic Division

Synthetic division is a simplified method of dividing polynomials, specifically when dividing by a linear factor of the form \(x - c\). This process is quicker than regular long polynomial division and is particularly useful when testing possible zeros or factoring.

• First, list the coefficients of the polynomial in descending order of power.
• Write the zero that you are testing (in this case, \(x = 1\)) to the left of the synthetic division bracket.
• Start by bringing down the first coefficient. Multiply this coefficient by the zero and add this product to the next coefficient. Repeat this process across all coefficients.
• The synthetic division yields a sequence of numbers. The last number in the sequence indicates the remainder. A remainder of zero confirms that \(x = 1\) is indeed a zero of the polynomial.
• The rest of the numbers form the coefficients of the reduced polynomial of one degree less.

Using synthetic division, you quickly see if the suspected zero works and simplify the polynomial into a possibly more manageable form.

Finding the Zero of a Polynomial

The zero of a polynomial refers to the value(s) of \(x\) at which the polynomial equals zero. For example, in the given polynomial \(p(x) = -x^{4}-x^{3}+18x^{2}+16x-32\), if substituting \(x = 1\) into \(p(x)\) results in 0, then \(x = 1\) is a zero.

• Substitute the given \(x\) value into the polynomial equation.
• If the result equals zero after simplifying, then the value is a zero.

Finding zeros is not just about plugging in numbers; it helps in breaking down complex polynomials into linear factors. These zeros are also called roots or solutions, and they play a key role in graphing the polynomial and understanding its behavior.

Factoring Polynomials

Factoring a polynomial means expressing it as a product of simpler polynomials. After establishing that \(x = 1\) is a zero, you use synthetic division to simplify the polynomial, uncovering factors step by step.

• Once you get the reduced polynomial from synthetic division, check if it can be factored further.
• If stopping at a quadratic or simpler form, test if standard factoring techniques apply, such as factoring by grouping or using the quadratic formula if necessary.
• If the polynomial is completely factored, it's expressed as a product of linear or irreducible quadratic factors.

Factoring is crucial as it reveals the polynomial's roots and simplifies solving polynomial equations that arise in various problems.

Quadratic Expressions

A quadratic expression is a polynomial of degree 2, typically written as \(ax^2 + bx + c\). Quadratics are central to polynomial division and factoring because when a polynomial is divided, it may reduce to a quadratic form that needs further factoring.
Once the cubic or higher degree polynomial is reduced through synthetic division, check if the leftovers are quadratic.

• For quadratics, use factoring techniques such as the quadratic formula, completing the square, or inspection methods.
• Each method helps in breaking down the quadratic into its simpler binomial form.

Quadratics are essential in polynomial division as they often represent the challenge of completely breaking down more complex polynomials into comprehensible units.