Problem 11 - 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. $$p(x)=x^{3}-5 x^{2}+8 x-4 ; x=2$$

Step-by-Step Solution

Answer

After the synthetic division, the polynomial factors as \(p(x) = (x-2)(x^{2}-3x+2)\). So, \(x = 2\) is indeed a zero of the polynomial.

Step 1: Substitute and Check

Substitute \(x = 2\) into the polynomial \(p(x) = x^{3}-5x^{2}+8x-4\). If the resultant value equals zero, then \(x = 2\) is a zero of the polynomial. So evaluate, \(p(2) = (2)^{3}-5(2)^{2}+8(2)-4\)

Step 2: Calculation

Calculate \(p(2) = 8 - 20 + 16 - 4 = 0\). Thus, \(x = 2\) is a zero of the polynomial.

Step 3: Factorization via Synthetic Division

To factorize the polynomial \(p(x)\), use synthetic division. First, set up the division with the zero \(2\) on the outside and the coefficients of \(p(x)\) inside. Synthetic division will give the quotient, which is another polynomial. This polynomial and the zero \(x - 2\), when multiplied will give the original polynomial.

Key Concepts

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a binomial of the form \(x - c\). It simplifies the division process by focusing on the coefficients of the polynomial, making it a quick way to check for zeros and to factor polynomials.

To perform synthetic division:

• Write down the coefficients of the polynomial, maintaining the order according to the power of \(x\). For \(p(x) = x^3 - 5x^2 + 8x - 4\), the coefficients are 1, -5, 8, and -4.
• Place the potential zero outside the division box. Here, it’s \(2\).
• Bring down the first coefficient (1) to the bottom row. Then multiply it by the number outside the division box (2 in this case) and add it to the next coefficient. Continue this process through all coefficients.

After completing synthetic division, you'll obtain a new set of coefficients, representing the quotient polynomial. If the remainder is zero, then \(x = c\) is indeed a zero of the original polynomial.

Polynomial Factorization

Factorization is the process of breaking down a polynomial into simpler components that can be multiplied to restore the original polynomial. For polynomials, these components are often found using their zeros.

Once you have determined a zero of the polynomial (like \(x = 2\)), you can factor the polynomial using this zero:

• Begin with the zero found from synthetic division, represented as \(x - 2\).
• Use the quotient obtained from synthetic division as the remaining factor. In our case, if synthetic division gives a quotient of \(x^2 - 3x + 2\), the original polynomial \(p(x)\) can be expressed as \((x - 2)(x^2 - 3x + 2)\).
• Further factor the quotient if possible. For instance, \(x^2 - 3x + 2\) can be factored to \((x-1)(x-2)\).

Therefore, the complete factorization of \(p(x) = x^3 - 5x^2 + 8x - 4\) becomes \((x - 2)^2(x - 1)\).

Cubic Polynomials

Cubic polynomials are polynomials of degree three. They generally take the form \(ax^3 + bx^2 + cx + d\). These polynomials can cross the x-axis up to three times, meaning they can have up to three real roots or zeros.

To handle cubic polynomials:

• Start by finding a root using simple substitution, as seen with \(x = 2\) for \(p(x)\).
• Utilizing synthetic division helps reduce the cubic equation into a quadratic one after identifying at least one root.
• Once a factor (such as \(x - 2\)) is extracted using synthetic division, you're left with a quadratic polynomial which can be solved using further factorization or the quadratic formula.

Understanding cubic polynomials involves recognizing patterns and using algebraic techniques such as synthetic division and factorization effectively to simplify the polynomial and find its zeros.