Functions such as linear, quadratic, square and cube functions all belong under one type function, the POYNOMIAL FUNCTION. This type of function is one of the foundations of other certain kinds of functions.
The study of graphs of polyommial functions is of considerable importance to scientists, astronomers, chemists, and physicists because the properties of these functions affect the behavior and shape of collected data in scientific research. For instance, technology has its foundation in the mathematical concepts of parabola and the quadratic formula. They have properties that are useful in making complex mechanisms that are useful in the advancement of the human race, such as satellite dishes, car headlights, radio telescopes, and reflecting telescopes.
Polynomial Functions of degree n is a function of the of the form:
f(x)=anxn+an-1xn-1+an-2xn-2+…a0, where n is a non-negative integer, and
an,an-1,…a0 are real numbers, and an not equal to 0.
The degree of polynomial functions is determined by the highest power of its terms.
The function h(x) =6x4+2x3+x2-x+4+7x5 has six terms.
The degree of terms are 5, 4, 3, 2, 1,and 0. Therefore the degree of this function is 5.
Methods in Finding Values of Polynomial Functions:
· The Remainder Theorem:
Example:
Find the remainder and value of P(x)=(x3-3x2+x+4)÷(x-2)
Solution:
Since x-c is represented by x-2 then, you must equate it to 0. Whereas:
x-c=0 x-2=0
x=c x=2 Therefore, substitute the x in the function as 2.
P(2)=(2)3-3(2)2+(2)+4
=8-12+2+4
=2 So, P(2)=2
· Synthetic Division:
Synthetic Division, hand-in-hand with the remainder theorem can be used as a convenient way to find values of polynomial functions or the remainder. In this kind of division the, coefficients are copied and the divisor is equated to zero.
Example:
P(x)=2x3-8x2+19x-12 divided by 2=x-3.
Solution:
First equate the divisor to 0 then find x.
x-3=0
x=3
Then copy the coefficients of the given function. With the value of x located at the upper left-hand corner of the solution.
3 2 -8 19 -12
Then, bring down the first numerical coefficient then multiply it to the value of x. Then, add the product to the next term. Repeat this process until to last term.
3 2 -8 19 -12
6 -6 39
2 -2 13 27
Therefore, the remainder of the function is 27 or can also written as P(3)=27.
· Factor Theorem:
It states that you can use either the Remainder Theorem or Synthetic Division to determine of the divisor is a factor of the equation or not. If it is a factor, then the function would be equal to 0. But if not, it would result to the remainder of the function.
Example:
When P(x)=x3-x2-4x+4 is divided by x-2.
Sol’n:
By Synthetic Division:
2 1 -1 -4 4
2 2 -4
1 1 -2 0
Or By Remainder Theorem: where x=2
P(x)=x3-x2-4x+4
P(2)=(2)3-(2)2-4(2)+4
=8-4-8+4
P(2)=0 Therefore, x-2 is a factor of the function P(x)=x3-x2-4x+4.
NOTE: The factors of the Function are also the zeros of that particular Function.
· Finding the Zeros of the Function:
As mentioned earlier, the factors of the function is also its zeros. But if it has a remainder, then the quadratic formula should be used to find its zeros.
Example:
Q(x)=x4-x3-11x2+9x+18 divided by (x+3), find its zeros.
By synthetic division:
-3 1 -1 -11 9 18
-3 12 -3 -18
1 -4 1 6 0
So, the quotient is x3-4x2+x+6 and x+3 is a factor of the function.
To find the other zeroes, factor the quotient, if not use the quadratic formula.
The study of graphs of polyommial functions is of considerable importance to scientists, astronomers, chemists, and physicists because the properties of these functions affect the behavior and shape of collected data in scientific research. For instance, technology has its foundation in the mathematical concepts of parabola and the quadratic formula. They have properties that are useful in making complex mechanisms that are useful in the advancement of the human race, such as satellite dishes, car headlights, radio telescopes, and reflecting telescopes.
Polynomial Functions of degree n is a function of the of the form:
f(x)=anxn+an-1xn-1+an-2xn-2+…a0, where n is a non-negative integer, and
an,an-1,…a0 are real numbers, and an not equal to 0.
The degree of polynomial functions is determined by the highest power of its terms.
The function h(x) =6x4+2x3+x2-x+4+7x5 has six terms.
The degree of terms are 5, 4, 3, 2, 1,and 0. Therefore the degree of this function is 5.
Methods in Finding Values of Polynomial Functions:
· The Remainder Theorem:
Example:
Find the remainder and value of P(x)=(x3-3x2+x+4)÷(x-2)
Solution:
Since x-c is represented by x-2 then, you must equate it to 0. Whereas:
x-c=0 x-2=0
x=c x=2 Therefore, substitute the x in the function as 2.
P(2)=(2)3-3(2)2+(2)+4
=8-12+2+4
=2 So, P(2)=2
· Synthetic Division:
Synthetic Division, hand-in-hand with the remainder theorem can be used as a convenient way to find values of polynomial functions or the remainder. In this kind of division the, coefficients are copied and the divisor is equated to zero.
Example:
P(x)=2x3-8x2+19x-12 divided by 2=x-3.
Solution:
First equate the divisor to 0 then find x.
x-3=0
x=3
Then copy the coefficients of the given function. With the value of x located at the upper left-hand corner of the solution.
3 2 -8 19 -12
Then, bring down the first numerical coefficient then multiply it to the value of x. Then, add the product to the next term. Repeat this process until to last term.
3 2 -8 19 -12
6 -6 39
2 -2 13 27
Therefore, the remainder of the function is 27 or can also written as P(3)=27.
· Factor Theorem:
It states that you can use either the Remainder Theorem or Synthetic Division to determine of the divisor is a factor of the equation or not. If it is a factor, then the function would be equal to 0. But if not, it would result to the remainder of the function.
Example:
When P(x)=x3-x2-4x+4 is divided by x-2.
Sol’n:
By Synthetic Division:
2 1 -1 -4 4
2 2 -4
1 1 -2 0
Or By Remainder Theorem: where x=2
P(x)=x3-x2-4x+4
P(2)=(2)3-(2)2-4(2)+4
=8-4-8+4
P(2)=0 Therefore, x-2 is a factor of the function P(x)=x3-x2-4x+4.
NOTE: The factors of the Function are also the zeros of that particular Function.
· Finding the Zeros of the Function:
As mentioned earlier, the factors of the function is also its zeros. But if it has a remainder, then the quadratic formula should be used to find its zeros.
Example:
Q(x)=x4-x3-11x2+9x+18 divided by (x+3), find its zeros.
By synthetic division:
-3 1 -1 -11 9 18
-3 12 -3 -18
1 -4 1 6 0
So, the quotient is x3-4x2+x+6 and x+3 is a factor of the function.
To find the other zeroes, factor the quotient, if not use the quadratic formula.
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