polynomials in action by lorence g. villaceran ateneo de zamboanga university
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POLYNOMIALS in
ACTIONby Lorence G. Villaceran
Ateneo de Zamboanga University
What is aPolynomial?
Polynomials in Action
A Polynomial is
• Is an algebraic expression which consist more than one summed term
• Is a finite sum of terms each of which is a real number or the product of a numerical factor and one or more factor raised to whole-number powers
• each part that is being added, is called a "term"
Polynomials in Action
An expression is not a Polynomial if
• It has a negative exponent
• It has a fractional exponent
• It has a variable in the denominator
• It has a variable inside the square root sign
Polynomials in Action
6x2
Determine the ff. if it is a polynomial or not
√x
Polynomial
1/x2 not a Polynomial
not a Polynomial
Polynomials in Action
4y6/3
9y3
Z-4
√x2
Polynomial
Polynomial
not a Polynomial
Polynomial
Polynomials in Action
TERM• It composes the polynomial
• It composes of a numerical, literal coefficient and exponent
Parts of a TERM
Numerical Coefficient
Literal Coefficient/Variable
6x2 Exponent/Degree
Polynomials in Action
Similar Terms• Terms that have the same degree
or exponent of the same variable
x2+xy-y2 2x2+3xy-2y2
Similar Term
Polynomials in Action
Types of PolynomialsMonomial
• If a polynomial contains only one term.
Binomial• If a polynomial contains two terms.
Trinomial• If a polynomial contains three terms.
Multinomial• If a polynomial contains more than three terms.
Polynomials in Action
Examples6x2
9y3+3y+4
x2+3x
x3+y-x+3
Binomial
Monomial
Trinomial
Multinomial
Polynomials in Action
Polynomials in Action
x3+x2y+3y3
x3y+wxy x3yz2
x3+x2y2+xy-y3
w3+wxy+x2z
Monomial
Multinomial
Binomial
Trinomial
Trinomial
Polynomials in Action
Four Fundamental Operations in Polynomial
Addition and Subtraction of Polynomials
Polynomials in Action
How to add polynomials in column form
• Arrange the polynomials in either descending or ascending order of the variable/s and place similar terms in same vertical column
• For addition, the similar terms by finding the sum of coefficients
• Apply rules in adding signed numbers and retain the common literal factor
Polynomials in Action
Add the following polynomials:
4x3+8x2-x-8; x2+6x+9; 9x3+5x-9
Example of adding polynomials in column form
4x3+8x2-x-8 x2+6x+9
9x3+ 5x-913x3+9x2+10x-8
Polynomials in Action
How to subtract polynomials in Column form
• Arrange the polynomials in either descending or ascending order of the variable/s and place similar terms in same vertical column
• For subtraction, set the subtrahend under the minuend so that similar terms fall in the same column
• Subtract the numerical coefficients of similar terms.
• Use the rule for subtraction for signed numbers and retain the common literal factor
Polynomials in Action
Subtract the following polynomials:
10y4-4y3-y2+y+20; 15y4-4y2-3y+7
Example of subtracting polynomials in Column form
10y4 - 4y3 - y2 + y + 20
15y4 4y2 3y 7- + + -+- -
- 5y4 – 4y3+ 3y2 + 4y + 13
Polynomials in Action
Multiplication of Polynomials
Polynomials in Action
Rules of ExponentLet a and b be the numerical coefficient or the literal coefficient and m, n and p be the exponent.
1. am x an = a(m+n)
2. (am)n = a(mxn)
3. (ab)m = ambm
4. (ambn)p = a(m x p)b(mxp)
5. am/an = a(m-n)
Polynomials in Action
Rules of ExponentLet a and b be the numerical coefficient or the
literal coefficient and m, n and p be the exponent.
6. a0 = 1
7. a1 = a
8. a-m = 1/am or 1/am = am/1 = am
9. am + am = 2am
10. am + an = am + an
Polynomials in Action
Example
am x an = a(m+n)
a3 x a2 = a3+2 = a5
(34)(37) = 311
(23a4)(25a6) = 28a10
(am)n = a(mxn)
(a5)3 = a5x3 = a15
(42)3 = 46
[(x2)2]2 = x2x2x2 = x8
Polynomials in Action
Example(ab)m = ambm
(ab)4 = a4b4
(5x)3 = 53x3
(4xy)5 = 45x5y5
(ambn)p = a(mxp)b(mxp)
(a2b3)4 = a2x4b3x4 = a8b12
(43x7y4)5 = 415x35y20
Polynomials in Action
Example
am/an = a(m-n)
a5/a3 = a5-3 = a2
a7/a10 = a7-10 = a3 or 1/a3
a3b8c12/a5b8c7 = a-2c5
a0 = 1
® 80 = 1
® 5a0 = 5(1) = 5
Polynomials in Action
Example
a1 = a
⌂ 81 = 8
⌂ 5a1 = 5(a) = 5a
a-m = 1/am or 1/am = am/1 = am
∂ a-3 = 1/a3
∂ a-5/b-2 = b2/a5
∂ 6a-2b5/7c- 6d3 = 6b5c6/7a2d3
Polynomials in Action
Exampleam + am = 2am
o a3 + a3 = 2a3
o 5a4 + 2a4 = 7a4
o 7a6 - 4a6 = 3a6
am + an = am + an
• a6 + a4 = a6 + a4
• 6a4 + 3a2 - 8a3 = 6a4 + 3a2 - 8a3
Polynomials in Action
Rules for multiplication of monomials
• Multiplying the coefficients by following the rule for multiplication of signed numbers to get the coefficient of the product
• Multiply the literal coefficients by following the laws of exponents to obtain the literal coefficient of the product
Polynomials in Action
Example of multiplying monomial by a monomial
Simplify (5x2)(–2x3)
Polynomials in Action
(5x2)(–2x3) = (5)(-2)(x2+3) = -10x5
Simplify (-3y5)(–9y0)
(-3y5)(–9y0) = (-3)(-9)(y5+0) = 27y5
Rules for multiplication of a polynomial by a monomials
• Apply the distributive property of multiplication over addition or subtraction
Polynomials in Action
Example of Multiplying monomial by a polynomial
Multiply 3x2 and 12x3-4x2
Polynomials in Action
= 3x2(12x3-4x2)
= 3x2(12x3) - 3x2(4x2)
= 3(12)(x2+3) - 3(4)(x2+2)
= 36x5-12x4
Example of Multiplying monomial by a polynomial
Multiply 7y4 and 5y4-9y3+8
Polynomials in Action
= 7y4(5y4-9y3+8)
= 7y4(5y4)-7y4(9y3)+7y4(8)
= 7(5)(y4+4)-7(9)(y4+3)+7(8)(y4)
= 35y8-63y7+56y4
• Take one term of the multiplier at a time and multiply the multiplicand
• Combine similar terms to get the required product
• Arrange the terms in descending order
Rules for multiplication of a polynomial by another polynomial
Polynomials in Action
Example of Multiplying polynomial by a polynomial
Multiply (3x+5) and (3x-4)
Polynomials in Action
= (3x)(3x)+(3x)(-4)+(5)(3x)+(5)(-4)
= 3(3)(x1+1)+(3)(-4)(x)+(5)(3)(x)+(5)(-4)
=9x2-12x+15x-20
=9x2+3x-20
Example of Multiplying polynomial by a polynomial
Multiply(2x2+3x+5) and (x2-2x-3)
Polynomials in Action
2x2+ 3x+ 5 x2- 2x- 3
-6x2- 9x-15 -2x3-6x2-10x
2x4+3x3+5x2
2x4+ x3 -7x2 -19x-15
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