### Presentation

```Sarah Byom and Samantha Kingery
singaporeolevelmaths.com
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If a polynomial f(x) is divided by x-k, then the
remainder is r=f(k)
Example: Find the remainder when
f(x)=3x2+7x-20 is divided by x-2.
K=2 (plug 2 in for x), so r=f(2)=3(2)2+7(2)-20=12+14-20=6.
The remainder is 6.
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Example: Find the remainder when
f(x) = x3-x2+4x+10 is divided by x-3
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F(x) has a factor x-k if and only if f(k)=0.
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Example: Is x-7 a factor of x2-10x+21?
Test: (7)2-10(7)+21 = 0.
Yes, the factors are (x-7) and (x-3).
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Use long division to find the quotient and
remainder when 3x3+5x2+8x+7 is divided by
3x+2.
1x2+1x+2
3x+2 )3x2+5x2+8x+7
3x2+2x2
3x2+8x+7
3x2+2x
6x+7
6x+4
3
Quotient
Dividend
Multiply:
1x2(3x+2)
Subtract
Multiply:
1x(3x+2)
Subtract
Multiply:
2(3x+2)
Remainder
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Example: Find the quotient and remainder
when x3-x2+4x+10 is divided by x-3.
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Synthetic division is the shortcut method for
the division of a polynomial by a linear divisor
x-k.
Example: Divide 2x3-3x2-5x-12 by x-3.
Zero of divisor 3
Line for products
Line for sums
1.
2.
3.
2 -3 -5 -12 Dividend
6 9 12
2 3
4 0 Remainder
Quotient
Multiply the zero of divisor (3) by the first coefficient of the dividend (2).
Write the product about the line and one column to the right.
Add the next coefficient of the dividend to the product just found and
record the sum below the line in the same column.
Repeat the “multiply” and “add” steps until the last row is complete.
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Divide 3x3-2x2+x-5 by x-1 using synthetic
division.
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Find all of the real zeros of f(x) =2x2-7x3-8x2+14x+8.
Use the Rational Zeros Theorem:
Factors of 8 : +1, +2, +4, +8
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=
+1, +2, +4, +8, +1/2
Factors of 2:
+1, +2
Graph the function, and compare the x-intercepts with the list of zeros.
Then use synthetic division with the possible zeros, which in this case would be 4 and
-1/2.
4
2 -7 -8 14 8
-1/2
2 1 -4 -2
8
4 -16 -8
-1
0
2
2
1 -4 -2
0
2 0 -4
0
Factor
(x-4)(x+1/2)(2x2-4)
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Write the function as a product of linear and