Question

show that the given value of x is a zero of the polynomial. use the zero to completely factor the polynomial.

1. \\(p(x) = x^3 - 5x^2 + 8x - 4\\); \\(x = 2\\)

Explanation:

Verify the given zero

Evaluate the polynomial \(p(x) = x^3 - 5x^2 + 8x - 4\) at \(x = 2\):

$$
LATEXBLOCK0
$$

Since \(p(2) = 0\), \(x = 2\) is a zero.

Perform polynomial division

Divide \(p(x)\) by \((x - 2)\) using synthetic division:

$$
LATEXBLOCK1
$$

The quotient is \(q(x) = x^2 - 3x + 2\).

Factor the quotient

Factor the quadratic polynomial \(x^2 - 3x + 2\):

$$ x^2 - 3x + 2 = (x - 1)(x - 2) $$

Write the complete factorization

Combine all factors to write the completely factored form:

$$
LATEXBLOCK2
$$

Answer:

\(p(2) = 0\), showing that \(x = 2\) is a zero.
The completely factored polynomial is:

$$p(x) = (x - 1)(x - 2)^2$$