Finding Zeros Of Polynomial P(x) = 3x^4 - 7x^3 - 37x^2 - 17x + 10

JU07/17/2025 08, 2025 by THE IDEN 66 views

Exploring Polynomial Zeros A Comprehensive Guide to P(x) = 3x^4 - 7x^3 - 37x^2 - 17x + 10

In the realm of mathematics, polynomials play a crucial role, and understanding their zeros is fundamental. This article delves into the intricacies of finding zeros for a given polynomial, specifically focusing on the polynomial P(x) = 3x^4 - 7x^3 - 37x^2 - 17x + 10. We will demonstrate how to verify given values as zeros and explore methods to uncover all other zeros. This comprehensive guide aims to provide a clear and detailed explanation for students, educators, and anyone interested in polynomial functions.

Verifying Zeros of P(x)

To begin, we are given two values, c = -2 and c = 1/3, and tasked with showing that these are zeros of the polynomial P(x). A zero of a polynomial is a value of x that makes the polynomial equal to zero. In other words, if P(c) = 0, then c is a zero of P(x). We will substitute these values into the polynomial and evaluate.

Verifying c = -2

Let’s substitute x = -2 into the polynomial P(x) = 3x^4 - 7x^3 - 37x^2 - 17x + 10:

P(-2) = 3(-2)^4 - 7(-2)^3 - 37(-2)^2 - 17(-2) + 10

First, we calculate the powers of -2:

(-2)^4 = 16
(-2)^3 = -8
(-2)^2 = 4

Now, substitute these values back into the expression:

P(-2) = 3(16) - 7(-8) - 37(4) - 17(-2) + 10 P(-2) = 48 + 56 - 148 + 34 + 10 P(-2) = 148 - 148 P(-2) = 0

Since P(-2) = 0, we have verified that c = -2 is indeed a zero of the polynomial P(x). This demonstrates that when x is -2, the polynomial evaluates to zero, confirming its status as a root or zero of the polynomial.

Verifying c = 1/3

Next, we will verify c = 1/3 as a zero of P(x). Substitute x = 1/3 into the polynomial:

P(1/3) = 3(1/3)^4 - 7(1/3)^3 - 37(1/3)^2 - 17(1/3) + 10

Calculate the powers of 1/3:

(1/3)^4 = 1/81
(1/3)^3 = 1/27
(1/3)^2 = 1/9

Substitute these values back into the expression:

P(1/3) = 3(1/81) - 7(1/27) - 37(1/9) - 17(1/3) + 10 P(1/3) = 1/27 - 7/27 - 37/9 - 17/3 + 10

To simplify, find a common denominator, which is 27:

P(1/3) = 1/27 - 7/27 - (37 * 3)/27 - (17 * 9)/27 + (10 * 27)/27 P(1/3) = (1 - 7 - 111 - 153 + 270)/27 P(1/3) = (271 - 271)/27 P(1/3) = 0/27 P(1/3) = 0

Since P(1/3) = 0, we confirm that c = 1/3 is also a zero of the polynomial P(x). This verification process is crucial in understanding the nature of polynomial functions and their roots.

Finding Other Zeros of P(x)

Now that we have verified that c = -2 and c = 1/3 are zeros of the polynomial P(x) = 3x^4 - 7x^3 - 37x^2 - 17x + 10, we can proceed to find the other zeros. Since P(x) is a fourth-degree polynomial, it has four zeros in total (counting multiplicity). We have found two, so we need to find the remaining two zeros.

Using Synthetic Division

Synthetic division is an efficient method for dividing a polynomial by a linear factor. Since -2 and 1/3 are zeros, (x + 2) and (x - 1/3) are factors of P(x). We will use synthetic division to divide P(x) by these factors sequentially.

Synthetic Division with c = -2

Set up the synthetic division with -2 as the divisor and the coefficients of P(x) as the dividend:

-2 | 3 -7 -37 -17 10
 | -6 26 22 -10
 ------------------------
 3 -13 -11 5 0

The result of the division is the quotient 3x^3 - 13x^2 - 11x + 5. The remainder is 0, which confirms that -2 is a zero.

Synthetic Division with c = 1/3

Now, we will divide the quotient 3x^3 - 13x^2 - 11x + 5 by (x - 1/3) using synthetic division:

1/3 | 3 -13 -11 5
 | 1 -4 -5
 ----------------
 3 -12 -15 0

The result of this division is the quotient 3x^2 - 12x - 15. The remainder is 0, confirming that 1/3 is a zero.

Solving the Quadratic Equation

After performing synthetic division twice, we are left with the quadratic equation:

3x^2 - 12x - 15 = 0

To find the remaining zeros, we need to solve this quadratic equation. First, we can simplify the equation by dividing all terms by 3:

x^2 - 4x - 5 = 0

Now, we can factor the quadratic:

(x - 5)(x + 1) = 0

Setting each factor equal to zero gives us the solutions:

x - 5 = 0 => x = 5 x + 1 = 0 => x = -1

Thus, the other two zeros of P(x) are 5 and -1.

Complete Set of Zeros

Combining the zeros we found, the complete set of zeros for the polynomial P(x) = 3x^4 - 7x^3 - 37x^2 - 17x + 10 is:

-2
1/3
5
-1

These are the four values of x that make the polynomial equal to zero. Understanding how to find these zeros is a fundamental skill in polynomial algebra and has various applications in mathematics and other fields.

Conclusion

In summary, we have demonstrated how to verify given values as zeros of a polynomial and how to find the remaining zeros using synthetic division and solving the resulting quadratic equation. For the polynomial P(x) = 3x^4 - 7x^3 - 37x^2 - 17x + 10, we verified that c = -2 and c = 1/3 are zeros. Subsequently, we used synthetic division to reduce the polynomial to a quadratic equation and found the other zeros to be 5 and -1. The complete set of zeros for P(x) is therefore {-2, 1/3, 5, -1}.

This process highlights the importance of understanding polynomial functions and their zeros, which are essential concepts in algebra and calculus. By mastering these techniques, students and enthusiasts can confidently tackle more complex polynomial problems and applications.