Given polynomial function f and a zero of f, find the other zeroes.
f(X)=4x^3-25x^2-154x+40;10
if 10 is a root (zero) , then (x - 10) is a factor
dividing by the factor gives ... 4 x^2 + 15 x - 4
factor the quotient to find the other two factors of f(X)
To find the other zeroes of the polynomial function f(x) = 4x^3 - 25x^2 - 154x + 40, given that 10 is a zero, we can use synthetic division or long division to divide f(x) by (x - 10).
Let's use synthetic division:
10 | 4 - 25 - 154 + 40
40 + 150 + 40
------------------------------------
4 + 15 - 4 + 80
The result after synthetic division is: 4x^2 + 15x - 4 + 80/(x - 10).
Now we have the new polynomial: f(x) = 4x^2 + 15x - 4 + 80/(x - 10).
To find the other zeroes, we need to solve the equation 4x^2 + 15x - 4 = 0. We can use factoring or the quadratic formula.
Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a), where a = 4, b = 15, and c = -4.
x = (-15 ± √(15^2 - 4(4)(-4))) / (2 * 4)
x = (-15 ± √(225 + 64)) / 8
x = (-15 ± √289) / 8
x = (-15 ± 17) / 8
So the other zeroes are:
x = (-15 + 17) / 8 = 2/8 = 1/4
x = (-15 - 17) / 8 = -32/8 = -4
Therefore, the other zeroes of the function f(x) = 4x^3 - 25x^2 - 154x + 40, given that 10 is a zero, are 1/4 and -4.
To find the other zeroes of the polynomial function f(x) = 4x^3 - 25x^2 - 154x + 40, given that one of the zeroes is 10, we can use polynomial long division or synthetic division.
Method 1: Synthetic Division
1. Set up the synthetic division as follows:
10 | 4 -25 -154 40
+______
2. Bring down the leading coefficient, which is 4.
10 | 4 -25 -154 40
+______
4
3. Multiply the divisor (10) by the first number, 4, in the dividend and write the result below the second number, which is -25.
10 | 4 -25 -154 40
+______
4
40
4. Add the numbers in the second column (-25 + 40) and write the result below.
10 | 4 -25 -154 40
+______
4
40
15
5. Multiply the divisor (10) by the new number (-25 + 40), which is 15, and write the result below the third number, which is -154.
10 | 4 -25 -154 40
+______
4
40
15
150
6. Add the numbers in the third column (-154 + 150) and write the result below.
10 | 4 -25 -154 40
+______
4
40
15
150
-4
7. Multiply the divisor (10) by the new number (-154 + 150), which is -4, and write the result below the fourth number, which is 40.
10 | 4 -25 -154 40
+______
4
40
15
150
-4
-40
8. Add the numbers in the fourth column (40 + -40). The result should be zero, confirming that 10 is a zero of the function.
10 | 4 -25 -154 40
+______
4
40
15
150
-4
-40
0
Now we have a quadratic function, which is 4x^2 + 15x - 4. To find the other zeroes, we can use factoring, the quadratic formula, or completing the square.
Method 2: Factoring the Quadratic Function
1. Factor the quadratic function 4x^2 + 15x - 4.
To find two numbers that multiply to -16 (4 * -4) and add up to 15, we can use trial and error or the quadratic formula. The factored form is:
(4x + 16)(x - 1)
2. Set each factor equal to zero and solve for x.
4x + 16 = 0
4x = -16
x = -4
x - 1 = 0
x = 1
The other zeroes of the polynomial function f(x) = 4x^3 - 25x^2 - 154x + 40 are -4 and 1, in addition to the given zero, 10.