We want to find the zeros of this polynomial:
[tex]p(x)=\left(x^2+4 x+3\right)\left(x^2-4\right)[/tex]
Plot all the zeros ([tex]x[/tex]-intercepts) of the polynomial in the interactive graph.
Answer (1)
Factor the quadratic x 2 + 4 x + 3 into ( x + 1 ) ( x + 3 ) .
Factor the quadratic x 2 − 4 into ( x − 2 ) ( x + 2 ) .
Set each factor to zero: x + 1 = 0 , x + 3 = 0 , x − 2 = 0 , x + 2 = 0 .
Solve for x to find the zeros: − 3 , − 2 , − 1 , 2 .
Explanation
Understanding the Problem We are given the polynomial p ( x ) = ( x 2 + 4 x + 3 ) ( x 2 − 4 ) and we want to find its zeros, which are the x -values where p ( x ) = 0 . These zeros correspond to the x -intercepts of the polynomial's graph. To find the zeros, we need to factor the polynomial completely.
Factoring the First Quadratic First, let's factor the quadratic x 2 + 4 x + 3 . We are looking for two numbers that multiply to 3 and add to 4. These numbers are 1 and 3. Therefore, x 2 + 4 x + 3 = ( x + 1 ) ( x + 3 ) .
Factoring the Second Quadratic Next, let's factor the quadratic x 2 − 4 . This is a difference of squares, so x 2 − 4 = ( x − 2 ) ( x + 2 ) .
Complete Factorization Now we can write the polynomial as p ( x ) = ( x + 1 ) ( x + 3 ) ( x − 2 ) ( x + 2 ) .
Finding the Zeros To find the zeros, we set each factor equal to zero and solve for x :
x + 1 = 0 ⟹ x = − 1
x + 3 = 0 ⟹ x = − 3
x − 2 = 0 ⟹ x = 2
x + 2 = 0 ⟹ x = − 2
So the zeros of the polynomial are x = − 1 , − 3 , 2 , − 2 .
Final Answer The zeros of the polynomial p ( x ) = ( x 2 + 4 x + 3 ) ( x 2 − 4 ) are x = − 3 , − 2 , − 1 , 2 . These are the x -intercepts of the graph of the polynomial.
Examples
Understanding polynomial zeros is crucial in many fields. For example, in engineering, zeros of a transfer function can represent stable operating points of a system. In economics, finding the roots of a cost function can help determine break-even points. Moreover, in computer graphics, polynomial roots can be used to model curves and surfaces. By finding the zeros, we can analyze and predict the behavior of various systems and models.
