Question

What is the corresponding binomial factor of a polynomial, \(P(x),\) given the value of the zero? a) \(P(1)=0\) b) \(P(-3)=0\) c) \(P(4)=0\) d) \(P(a)=0\)

Short answer

(a) (x-1), (b) (x+3), (c) (x-4), (d) (x-a)

Step-by-step solution

- Understanding the Concept

The polynomial's factors are derived from its zeros. This means that if you know a zero of the polynomial, you can determine a corresponding binomial factor. The binomial factor will be of the form \(x-c\), where \(c\) is the zero.

- Apply the Concept to Example a

Given \(P(1)=0\), the zero is \(1\). Therefore, the corresponding binomial factor is \((x-1)\).

- Apply the Concept to Example b

Given \(P(-3)=0\), the zero is \(-3\). Therefore, the corresponding binomial factor is \((x+3)\).

- Apply the Concept to Example c

Given \(P(4)=0\), the zero is \(4\). Therefore, the corresponding binomial factor is \((x-4)\).

- Apply the Concept to Example d

Given \(P(a)=0\), the zero is \(a\). Therefore, the corresponding binomial factor is \((x-a)\).

Key concepts

binomial factor

Understanding binomial factors is essential when working with polynomials. A binomial factor is a simple algebraic expression of the form \(x - c\). Here, \(c\) represents a constant. The roots or zeros of the polynomial directly relate to these binomial factors. If we know a zero, we can determine its corresponding binomial factor. For example, if the zero is \(3\), the binomial factor will be \(x - 3\). This relationship helps in factoring polynomials and finding their solutions.

zeros of polynomial

The zeros of a polynomial are the values of \(x\) for which the polynomial equals zero. Essentially, these are the roots of the polynomial where the graph of the function intersects the x-axis. To find a zero, we set the polynomial equal to zero and solve for \(x\). For instance, if \(P(2) = 0\), then \(2\) is a zero of the polynomial \(P(x)\).
Identifying zeros is crucial because they help us determine the binomial factors. For example, knowing that \(2\) is a zero means we can write the factor as \(x - 2\). Each zero provides a clue to the polynomial’s factors and structure.

factoring polynomials

Factoring polynomials involves breaking down a polynomial into simpler components (its factors) that when multiplied together give back the original polynomial. This often starts with identifying the polynomial's zeros. Each zero corresponds to a binomial factor, making them essential in the factoring process.
To factor a polynomial, follow these steps:

• Find the zeros of the polynomial.
• For each zero \(c\), write down the binomial factor \(x - c\).
• Multiply all identified binomial factors together.

For example, if the polynomial \(P(x) = x^2 - 5x + 6\) has zeros \(2\) and \(3\), we get binomial factors \(x-2\) and \(x-3\). The factored form will be \(P(x) = (x-2)(x-3)\).

roots of polynomial

Roots of a polynomial, also known as zeros, are the solutions to the polynomial equation \(P(x) = 0\). They are the values of \(x\) for which the polynomial produces zero. Finding the roots involves solving the polynomial equation. These roots have a direct relationship with binomial factors.
For example, if \(x = 4\) is a root, it tells us that \(4\) is one of the solutions to \(P(x) = 0\) and corresponds to the binomial factor \(x - 4\). Knowing the roots allows us to decompose the polynomial into its factors, which can simplify many algebraic operations and help in solving complex equations.