Page 154 - Algebra

8.3. Factoring by Grouping and Finding Zeros
Factoring by grouping
Polynomial functions with four terms can be solved by grouping the appropriate terms. You need to
factor out the binomial common to both the terms.
For example,
5x3 –20x2 +x–4=5x2(x–4)+1(x–4)=(5x2 –1)(x–4)
Zeros of a polynomial
Zeros of a polynomial are the points where the polynomial becomes equal to zero. Also, the zeros correspond to the x-intercept of the graph of the polynomial.
REMEMBER:
For f(x) = anxn + an-1xn-1 +........ao where ao, a1.....an
‘b’ is a zero of a polynomial function and all the following equation holds true:
• • •
x = b is a root or solution of the equation f(x) = 0
The graph of the polynomial function intersects the x-axis at (b, 0)
x – b is a factor of the equation f(x) = 0
Worked Example
Find the real root of f(x) = 5x3 – 20x2 + 8x – 32
Solution:
f(x)=5x3 –20x2 +8x–32
To find the solution, equate f(x) = 0 5x3 –20x2 +8x–32=0
Group associated terms 5x3 –20x2 +8x–32=0 5x2(x – 4) + 8(x – 4) = 0 (5x2 +8)(x–4)=0
x = 4 or 5x2 = – 8
5x2 = – 8 will not result in real roots.
Therefore, the zeros or the real root of the equation is 4.
8.4. Remainder and Factor Theorem of Polynomials
Remainder Theorem
When a polynomial f(x) is divided by x –a, the remainder is f(a).
Page 153 of 177
Algebra I & II