ZEROS=ROOTS=SOLUTIONS Equals x intercepts. Another Memory slide.
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Transcript of ZEROS=ROOTS=SOLUTIONS Equals x intercepts. Another Memory slide.
ZEROS=ROOTS=SOLUTIONSEquals x intercepts
Another Memory slide
Looking for two numbers that add to column A and Multiply to column B
• 7• 6• 8• -3• -5• 13• 1• -4
• 12• 8• -9• -10• 6• 30• -2• 4
3 and 4
4 and 29 and -1-5 and 2-3and -23 and 102 and -1-2 and -2
A B
PolynomialsDefining Polynomials
Adding Like Terms
• Monomials - a number, a variable, or a product of a number and one or more variables. 4x, 20x2yw3, -3, a2b3, and 3yz are all monomials.
• Polynomials – one or more monomials added or subtracted
• 4x + 6x2, 20xy - 4, and 3a2 - 5a + 4 are all polynomials.
Vocabulary
Like TermsLike Terms
Like Terms refers to monomials that have the same variable(s) but may have different coefficients. The variables in the terms must have the same powers.
Which terms are like? 3a2b, 4ab2, 3ab, -5ab2
4ab2 and -5ab2 are like.
Even though the others have the same variables, the exponents are not the same.
3a2b = 3aab, which is different from 4ab2 = 4abb.
Like TermsLike Terms
Constants are like terms.
Which terms are like? 2x, -3, 5b, 0
-3 and 0 are like.
Which terms are like? 3x, 2x2, 4, x
3x and x are like.
Which terms are like? 2wx, w, 3x, 4xw
2wx and 4xw are like.
Add: (x2 + 3x + 1) + (4x2 +5)
Step 1: Underline like terms:
Step 2: Add the coefficients of like terms, do not change the powers of the variables:
Adding PolynomialsAdding Polynomials
(x2 + 3x + 1) + (4x2 +5)
Notice: ‘3x’ doesn’t have a like term.
(x2 + 4x2) + 3x + (1 + 5)
5x2 + 3x + 6
Adding PolynomialsAdding Polynomials
1) 3x3 7x 3x3 4x 6x3 3x
2) 2w2 w 5 4w2 7w 1 6w2 8w 4
3) 2a3 3a2 5a a3 4a 3 3a3 3a2 9a 3
• Add the following polynomials; you may stack them if you prefer:
Subtract: (3x2 + 2x + 7) - (x2 + x + 4)
Subtracting PolynomialsSubtracting Polynomials
Step 1: Change subtraction to addition (Keep-Change-Change.).
Step 2: Underline OR line up the like terms and add.
(3x2 + 2x + 7) + (- x2 + - x + - 4)
(3x2 + 2x + 7)
+ (- x2 + - x + - 4)
2x2 + x + 3
Subtracting PolynomialsSubtracting Polynomials
1) x2 x 4 3x 2 4x 1 2x2 3x 5
2) 9y2 3y 1 2y2 y 9 7y2 4y 10
3) 2g2 g 9 g3 3g2 3 g3 g2 g 12
• Subtract the following polynomials by changing to addition (Keep-Change-Change.), then add:
WHEN MULTIPLYING LIKE BASES, YOU ADD THE EXPONENTS
FOR EXAMPLE:
NOW YOU TRY:
x2 -5x +4
2x
-5
5) Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH
binomials. You must use the distributive property or box method.
2x3
-5x2
-10x2
+25x
+8x
-20
Almost done!
Go to the next slide!
x2 -5x +4
2x
-5
5) Multiply (2x - 5)(x2 - 5x + 4) Combine like terms!
2x3
-5x2
-10x2
+25x
+8x
-20
2x3 – 15x2 + 33x - 20
#1 Divide a polynomial by a monomial.
2
564 3 xx
2
5
2
6
2
4 3
xx
2
532 3 xx
WHEN DIVIDING LIKE BASES, YOU SUBTRACT THE EXPONENTS.
FOR EXAMPLE:
NOW YOU TRY:
#2 Divide a polynomial by a monomial.
x
xxx
2
462 23
x
x
x
x
x
x
2
4
2
6
2
2 23
232 xx
#3 Divide the following numbers.
13
5733
5733134
5253
4
5213
1 Check
13
441
5733
130
Long Division
1. Divide the first term of the numerator by the first term of the denominator.
2. Distribute your quotient by the entire polynomial in the denominator.
3. Distribute a negative by through your product.
#4 Divide the following numbers
(Long Division)11
8034
8034117
7733
3
3304
0
04 Remainder
11
4730
#5 Dividing a polynomial by a polynomial (Long Division)
2
822
x
xx
822 2 xxxx
xx 22 x4 8
4
84 x0
Check
)2)(4( xx
8422 xxx
822 xx
x
x2
)2(xx
x
x4 )2(4 x
POLYNOMIALS – DIVIDINGEX – Long division
• (5x³ -13x² +10x -8) / (x-2)
5x³ - 13x² + 10x - 8x - 2
5x²
5x³ - 10x²- ( )
-3x² + 10x
- 3x
-3x² + 6x- ( )
4x - 8
4x - 8- ( )
+ 4
0
R 0
36x
#6 Dividing a polynomial by a polynomial (Long Division)
7
42133
36
x
xx
3x
36 7xx 42
6
426 3 x0
Check
)6)(7( 33 xx
4276 336 xxx
4213 36 xx
)7( 33 xx
)7(6 3x
42137 363 xxx
3
6
x
x
3
36
x
x
#7 Dividing a polynomial by a polynomial (Long Division)
1
124
x
xx
1001 234 xxxxx
3x
34 xx 3x 2x
2x
23 xx 22x x0
x2 x2
xx 22 2 x2 1
2
12 x22 x3
1
3
x
#8 Dividing a polynomial by a polynomial (Long Division)
yy
yyy
523
19730152
23
7301915253 232 yyyyyy5
yyy 102515 23 26y y20
2
4106 2 yy11y10
7
253
11102
yy
y
