Transcript Polynomials - Mr

Polynomials
Higher Maths
Polynomials introduction
Polynomials 1
Factors
Ans
Polynomials
Ans
Curves cutting the x and the y axis
Ans
Quotient and remainder
Factors of the form ax + b
Ans
Polynomial problems 1
Click on a topic
Polynomial problems 2
Polynomial exam level questions
Past Paper questions
Ans
Polynomials (Introduction)
Use nested multiplication to find the values of the functions below:
1. f(x) = x3 + 2x2 + 3x + 5 , find f(3)
2. f(x) = 3x2 - 4x + 7 , find f(2)
3. f(x) = 2x3 - x2 – x - 1 , find f(-2)
4. f(x) = x4 + 2x3 – x2 + x + 1 , find f(3)
5. f(x) = x3 - 2x2 - 3x - 7 , find f(1)
6. f(x) = x4 - x3 + x2 + x + 2 , find f(4)
7. f(x) = x4 - 2x2 - 2 , find f(-1)
8. f(x) = 2x5 + x3 - 6x + 8 , find f(2)
9. f(x) = 7x2 - 2x + 3 , find f(-1)
10. f(x) = x3 - 10x2 - 8 , find f(-2)
11. f(x) = 2x5 + 3x4 - x2 + 2x - 3 , find f(-1)
12. f(x) = x6 + 2x4 -3x3 + 2x2 + 1 , find f(-3)
Polynomials (1)
Use nested multiplication to find the values of the functions below
1
f(x) = 2x3 - 3x2 + 5x + 1 , find f(4)
2
f(x) = x4 - x3 + 2x2 + x + 3 , find f(-2)
3
f(x) = 3x3 - 4x2 + 5 , find f(2)
4
f(x) = 2x4 + 3x3 -8x + 5 , find f(3)
5
f(x) = 2x5 + 3x3 + 4x2 - 7x - 1 , find f(1)
6
f(x) = x4 - 6x3 + 3x2 + 4x + 2 , find f(-2)
7
f(x) = 4x3 - 7x2 - x - 2 , find f(-1)
8
f(x) = 2x4 + 4x3 - 6x + 8 , find f(1/2)
9
f(x) = 8x2 - 2x + 3 , find f(-1/2)
10
f(x) = x3 - 10x2 + 5x - 8 , find f(-2)
11
f(x) = 2x5 + 3x4 - x2 - 2x + 3 , find f(-1)
12
f(x) = x6 + 2x4 -3x3 + 2x2 + 1 , find f(-3)
Solutions on next slide
Solutions
1
f(4) = 101
2
f(-2) = 33
3
f(2) = 13
4
f(3) = 224
5
f(1) = 1
6
f(-2) = 70
7
f(-1) = -12
8
f(1/2) = 5.625
9
f(-1/2) = 6
10
f(-2) = -66
11
f(-1) = 5
12
f(-3) = 991
Factors
1
Show that x - 2 is a factor of x3 + x2 - 10x + 8 and hence
factorise fully.
2
Show that x - 4 is a factor of x3 - 4x2 - 9x + 36 and hence
factorise fully.
3
Show that x + 2 is a factor of x3 + 4x2 + x - 6 and hence
factorise fully.
4
Show that x + 1 is a factor of x3 - 6x2 + 3x + 10 and hence
factorise fully.
5
Show that x - 2 is a factor of 2x3 - 7x2 + 7x - 2 and hence
factorise fully.
6
Show that x + 4 is a factor of 3x3 + 14x2 + 7x - 4 and hence
factorise fully.
7
Show that x + 3 is a factor of x3 + 3x2 - 25x - 75 and hence
factorise fully.
8
Show that x - 3 is a factor of 4x3 - 21x2 + 29x - 6 and hence
factorise fully.
9
Show that x - 1 is a factor of 8x3 - 14x2 + 7x - 1 and hence
factorise fully.
Factors - Some Solutions
1
(x - 2)(x - 1)(x + 4)
2
(x + 3)(x - 3)(x - 4)
3
(x - 1)(x + 3)(x + 2)
4
(x + 1)(x - 2)(x - 5)
5
(x - 2)(2x - 1)(x - 1)
Curves cutting the x and y axes
In each example, find the points where the curve cuts the x and y axes.
1. y = x3 + x2 - 10x + 8
2. y = x3 + 6x2 + 11x + 6
3. y = x3 - 8x2 + 17x - 10
4. y = x3 - x2 - 10x - 8
5. y = x3 + 4x2 + x - 6
6. y = x3 + x2 - 16x + 20
7. y = 2x3 - 17x2 + 22x - 7
8. y = 6x3 - 17x2 + 11x - 2
9. y = 4x3 + 11x2 + 5x - 2
10. y = 3x3 - 5x2 - 4x + 4
11. y = x3 - 8x2 + 11x + 20
12. y = 2x3 - x2 - 15x + 18
13. y = x4 - 5x3 + 5x2 + 5x - 6
14. y = 2x4 + 9x3 + 6x2 - 11x - 6
Solutions on next slide
Curves cutting the x and y axes - Solutions
1. (0, 8) (-4, 0) (2, 0) (1, 0)
2. (0, 6) (-1, 0) (-2, 0) (-3, 0)
3. (0, -10) (1, 0) (2, 0) (5, 0)
4. (0, -8) (-2, 0) (-1, 0) (4, 0)
5. (0, -6) (1, 0) (-2, 0) (-3, 0)
6. (0, 20) (2, 0) (2, 0) (-5, 0)
7. (0, -7) (7, 0) (½, 0) (1, 0)
8. (0, -2) (2, 0) (½, 0) (1/3, 0)
9. (0, -2) (¼, 0) (-1, 0) (-2, 0)
10. (0, 4) (2/3, 0) (-1, 0) (2, 0)
11. (0, 20) (4, 0) (5, 0) (-1, 0)
12. (0, 18) (3/2, 0) (2, 0) (-3, 0)
13. (0, -6) (1,0) (2, 0) (3, 0) (-1, 0)
14. (0, -6) (-½, 0) (1, 0) (-2, 0) (-3, 0)
Quotient and Remainder
Find the quotient and the remainder in each example.
1
x2 – 5x + 2 ÷ (x – 3)
2
2x2 + x + 3 ÷ (x – 1)
3
x3 + x2 – 3x + 1 ÷ (x + 2)
4
x3 - 2x2 – x - 3 ÷ (x + 1)
5
x2 – x - 2 ÷ (x – 1)
6
2x2 - 3x - 4 ÷ (x + 3)
7
x3 - 4x2 – 4x + 16 ÷ (x - 4)
8
x2 – 6x - 7 ÷ (x – 7)
9
x2 + x + 5 ÷ (x – 2)
10
x3 + 2x2 – 4 ÷ (x - 1)
11
2x3 - 3x2 – 4x + 7 ÷ (x + 1)
12
x3 - 4x2 – 7x + 10 ÷ (x - 5)
13
x4 + x2 + 1 ÷ (x - 2)
14
x3 - x2 + x - 1 ÷ (x - 1)
Division by ax + b
Find the quotient and remainder in each of the following exercises.
1.
2.
3.
4.
5.
6.
7.
8.
9.
4x2 + 6x - 2 divided by 2x - 1
4x3 - 2x2 + 6x - 1 divided by 2x - 1
6x2 - 5x + 2 divided by 3x - 1
9x2 - 6x - 10 divided by 3x + 1
3x3 + 5x2 - 11x + 8 divided by 3x - 1
2x3 + 7x2 - 5x + 4 divided by 2x + 1
2x3 - x2 - 1 divided by 2x + 3
5x3 + 21x2 + 9x - 1 divided by 5x + 1
6x3 + x2 + 1 divided by 2x - 3
Solutions on next slide
Division by ax + b Solutions
1.
2.
3.
4.
5.
6.
7.
8.
Quotient
Remainder
2x + 4
2
2x2 + 3
2
2x - 1
1
3x - 3
-7
x2 + 2x - 3
5
x2 + 3x - 4
8
x2 - 2x + 3
-10
x2 + 4x + 1
-2
Polynomials Problems 1
1. Show that x-4 is a factor of 2x2 – 11x + 12 and hence factorize fully.
2. Factorize fully x3 – 11x2 + 26x – 16
3. If x+3 is a factor of x3 + kx2 + 7x + 3 , find k and hence factorize fully.
4. Show that x=2 is a root of the equation x3 + 5x2 - 4x – 20 = 0 and
find the other roots.
5. Find the points where the curve y = x3 + 10x2 - 9x – 90 cuts
the coordinate axis.
6. Factorize fully x3 + 2x2 - x – 2.
7. If x-1 is a factor of x3 - 3x2 + kx – 1, find k and hence factorize fully.
8. Show that x=1 is a root of the equation x3 - 9x2 + 20x–12 = 0
and find the other roots.
9. Show that x =-4 is a root of the equation 6x3 + 25x2 + 2x–8 = 0
and find the other roots.
10. If x-2 is a factor of f(x) = 2x3 + kx2 + 7x + 6 , find k and hence
solve the equation f(x) = 0 with this value of k.
11. The same remainder is obtained when x2 + 3x – 2 and
x3 - 4x2 + 5x + p are divided by x+1. Find p.
Polynomials problems 2
1. Find k if x+3 is a factor of x3 – 3x2 + kx + 6
2. Find p if x4 + 4x3 + px2 + 4x + 1 has x+1 as a factor.
Hence factorize fully.
3. If x+3 and x-1 are factors of f(x) = x4 + 2x3 - 7x2 + ax + b , find a and b
and hence factorize fully.
4. If x+2 is a factor of x3 + kx2 - x – 2 , find k
and hence factorize fully.
5. If x=3 is a root of the equation x3 – 37x + k = 0, find k
and hence find all the other roots.
6. Given that x-2 is a factor of f(x) = 2x3 + kx2 + 7x + 6, find k.
Hence solve the equation f(x) = 0 with this value of k.
7. Find k if 2x3 + x2 + kx – 8 is divisible by x+2.
8. Find k if x3 + kx2 - 6x + 8 has a factor x-4.
Hence factorize the expression fully.
Revision - Exam level questions
1. 2x + 1 is a factor of 2x3 – tx2 + x + 2. Find t.
2. If x + 1 and x - 3 are factors of f(x) = 2x3 - 5x2 + px + q, find p and q.
3. Given that 2x - 1 is a factor of 4x3- 4x2 + kx + 15 , find k.
Factorize fully when k has this value.
4. Find the points where the curve y = 4x3 – 4x2 - 29x + 15 cuts the x-axis.
5. Factorize fully
a) 2x3- 3x2 - 11x + 6
b) 3x3 - 2x2 - 19x - 6
6. x3 + kx2- 13x - 10 is divisible by x + 2. Find the value of k.
7. 2x3 - 9x2 + ax + 30 is divisible by 2x - 3. Find a.
8. x + 3 is a factor of 3x3+ 2x2 + nx + 6.
9.
Find n then factorize fully.
x4 - 2x3 + kx2 + 3x - 2 has x + 2 as a factor. Find the value of k.
10. Factorize fully x3+ 6x2+ 9x + 4 and hence solve
x3+ 6x2+ 9x + 4 = 0.Find the stationary points on the
curve y = x3+ 6x2+ 9x + 4 and determine their nature. Sketch
the curve.
11. If x - 1 and x + 3 are both factors of 2x3+ ax2 + bx + 3, find the
values of a and b.
12. Find k if x + 1 is a factor of x3 + kx2 - 5x - 6. Find the other
factors when k has this value.
13. Solve the equation x3- x2 + x - 6 = 0. Hence find the
equation of the tangent to the curve y = x3- x2 + x - 6 at the
points where it cuts the x-axis. Find the equation of the
tangent at the point where the curve crosses the y-axis.
Show that the two tangents meet at (3 /2, -9/2).
14. If f(x) = 3x4 + 8x3 – 6x2 , solve the equation f '(x) = 24.
Polynomials - (Questions from past papers)
1.
Factorise fully 2x3 – 3x2 - 11x + 6
2.
Factorise fully x3 – 6x2 + 9x – 4
3.
Factorise fully 2x3 + 5x2 - 4x – 3
4.
Find ‘p’ if (x+3) is a factor of x3 – x2 + px + 15.
5. When f(x) = 2x4 – x3 + px2 + qx + 12 is divided by (x – 2)
the remainder is 114. One factor of f(x) is (x + 1). Find p and q
6. One root of 2x3 – 3x2 + px + 30 = 0 is x = - 3.
Find ‘p’ and hence find the other roots.
7. Show that (x – 3) is a factor of f(x) = 2x3 + 3x2 – 23x – 12 and
hence factorise f(x) fully.
Continued on the next slide
8. Find a real root of the equation 2x3 – 3x2 + 2x – 8 = 0
Show algebraically that there are no other real roots.
9. Find ‘k’ if (x – 2) is a factor of x3 + kx2 – 4x – 12
and hence factorise fully.
10. Express x3 - 4x2 – 7x + 10 in fully factorised form.
11. Show that x = 2 is a root of the equation
2x3 + x2 – 13x + 6 = 0 and hence find the other roots.
12. Given that (x + 2) is a factor of 2x3 + x2 + kx + 2, find the value of k.
Hence solve the equation 2x3 + x2 + kx + 2 = 0 when k takes this value.
13. Given that (x – 2) and (x + 3) are factors
of f(x) = 3x3 + 2x2 + cx + d, find the values of ‘c’ and ‘d’.
Solutions on next slide
Answers to – Polynomials past paper questions
Question
Solution
1
(x – 3)(2x – 1)(x + 2)
2
(x – 1)(x – 1)(x - 4)
3
(x – 1)(2x + 1)(x + 3)
4
p = -7
5
4p + 2q = 78; p – q = -15 solve to give p=8 and q = 23
6
Roots are x=-3 , 2 and 5/2
7
(x – 3)(2x + 1)(x + 4)
8
(x-2)(2x2+x+4) = 0 So x = 2 is a root and there are no more roots
because you cannot take the square root of a negative number
which occurs when you apply the quadratic formula.
9
K = 3 ; So this gives f(x) = (x – 2)(x + 3)(x + 2)
10
(x – 1)(x –5)(x + 2)
11
Roots are x = 2 , x = ½ and x = -3
12
k = -5 ; x = -2 , ½ and 1
13
2c + d + 32 = 0, -3c + d -63 = 0 solve to
give c = -19 and d = 6