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Polynomial Results
Polynomial Results1. If P(x) has k distinct real zeros, , then;
is a factor of P(x).kaaaa ,,,, 321
kaxaxaxax 321
Polynomial Results1. If P(x) has k distinct real zeros, , then;
is a factor of P(x).kaaaa ,,,, 321
kaxaxaxax 321
e.g. Show that 1 and –2 are zeros of and hence factorise P(x).
4 3 2( ) 3 5 10P x x x x x
Polynomial Results1. If P(x) has k distinct real zeros, , then;
is a factor of P(x).kaaaa ,,,, 321
kaxaxaxax 321
e.g. Show that 1 and –2 are zeros of and hence factorise P(x).
4 3 2( ) 3 5 10P x x x x x
4 3 21 1 1 3 1 5 1 100
P
Polynomial Results1. If P(x) has k distinct real zeros, , then;
is a factor of P(x).kaaaa ,,,, 321
kaxaxaxax 321
e.g. Show that 1 and –2 are zeros of and hence factorise P(x).
4 3 2( ) 3 5 10P x x x x x
4 3 21 1 1 3 1 5 1 100
P
4 3 22 2 2 3 2 5 2 10
0P
Polynomial Results1. If P(x) has k distinct real zeros, , then;
is a factor of P(x).kaaaa ,,,, 321
kaxaxaxax 321
e.g. Show that 1 and –2 are zeros of and hence factorise P(x).
4 3 2( ) 3 5 10P x x x x x
4 3 21 1 1 3 1 5 1 100
P
4 3 22 2 2 3 2 5 2 10
0P
1, 2 are zeros of ( ) and 1 2 is a factorP x x x
Polynomial Results1. If P(x) has k distinct real zeros, , then;
is a factor of P(x).kaaaa ,,,, 321
kaxaxaxax 321
e.g. Show that 1 and –2 are zeros of and hence factorise P(x).
4 3 2( ) 3 5 10P x x x x x
4 3 21 1 1 3 1 5 1 100
P
4 3 22 2 2 3 2 5 2 10
0P
1, 2 are zeros of ( ) and 1 2 is a factorP x x x 4 3 2( ) 3 5 10P x x x x x
Polynomial Results1. If P(x) has k distinct real zeros, , then;
is a factor of P(x).kaaaa ,,,, 321
kaxaxaxax 321
e.g. Show that 1 and –2 are zeros of and hence factorise P(x).
4 3 2( ) 3 5 10P x x x x x
4 3 21 1 1 3 1 5 1 100
P
4 3 22 2 2 3 2 5 2 10
0P
1, 2 are zeros of ( ) and 1 2 is a factorP x x x 4 3 2( ) 3 5 10P x x x x x
2 2 x x
Polynomial Results1. If P(x) has k distinct real zeros, , then;
is a factor of P(x).kaaaa ,,,, 321
kaxaxaxax 321
e.g. Show that 1 and –2 are zeros of and hence factorise P(x).
4 3 2( ) 3 5 10P x x x x x
4 3 21 1 1 3 1 5 1 100
P
4 3 22 2 2 3 2 5 2 10
0P
1, 2 are zeros of ( ) and 1 2 is a factorP x x x 4 3 2( ) 3 5 10P x x x x x
2 2 x x 2x
Polynomial Results1. If P(x) has k distinct real zeros, , then;
is a factor of P(x).kaaaa ,,,, 321
kaxaxaxax 321
e.g. Show that 1 and –2 are zeros of and hence factorise P(x).
4 3 2( ) 3 5 10P x x x x x
4 3 21 1 1 3 1 5 1 100
P
4 3 22 2 2 3 2 5 2 10
0P
1, 2 are zeros of ( ) and 1 2 is a factorP x x x 4 3 2( ) 3 5 10P x x x x x
2 2 x x 2x 5
Polynomial Results1. If P(x) has k distinct real zeros, , then;
is a factor of P(x).kaaaa ,,,, 321
kaxaxaxax 321
e.g. Show that 1 and –2 are zeros of and hence factorise P(x).
4 3 2( ) 3 5 10P x x x x x
4 3 21 1 1 3 1 5 1 100
P
4 3 22 2 2 3 2 5 2 10
0P
1, 2 are zeros of ( ) and 1 2 is a factorP x x x 4 3 2( ) 3 5 10P x x x x x
2 2 x x 2x 5
21 2 5x x x
2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321
naaaa ,,,, 321
2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321
naaaa ,,,, 321
3. A polynomial of degree n cannot have more than n distinct real zeros
2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321
naaaa ,,,, 321
3. A polynomial of degree n cannot have more than n distinct real zeros
e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.Write down;
a) a possible polynomial
2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321
naaaa ,,,, 321
3. A polynomial of degree n cannot have more than n distinct real zeros
e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.Write down;
a) a possible polynomial
2( ) 2 7P x x x
2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321
naaaa ,,,, 321
3. A polynomial of degree n cannot have more than n distinct real zeros
e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.Write down;
a) a possible polynomial
2( ) 2 7P x x x ( )Q xwhere - Q(x) is a polynomial and does not have a zero at 2 or –7
2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321
naaaa ,,,, 321
3. A polynomial of degree n cannot have more than n distinct real zeros
e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.Write down;
a) a possible polynomial
2( ) 2 7P x x x ( )Q xwhere - Q(x) is a polynomial and does not have a zero at 2 or –7
k
- k is a real number
2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321
naaaa ,,,, 321
3. A polynomial of degree n cannot have more than n distinct real zeros
e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.Write down;
a) a possible polynomial
2( ) 2 7P x x x ( )Q xwhere - Q(x) is a polynomial and does not have a zero at 2 or –7
k
- k is a real number
b) a monic polynomial of degree 3.
2. If P(x) has degree n and has n distinct real zeros, , then naxaxaxaxxP 321
naaaa ,,,, 321
3. A polynomial of degree n cannot have more than n distinct real zeros
e.g. The polynomial P(x) has a double zero at –7 and a single zero at 2.Write down;
a) a possible polynomial
2( ) 2 7P x x x ( )Q xwhere - Q(x) is a polynomial and does not have a zero at 2 or –7
k
- k is a real number
b) a monic polynomial of degree 3.
2( ) 2 7P x x x
c) A monic polynomial of degree 4
c) A monic polynomial of degree 4
2( ) 2 7P x x x
c) A monic polynomial of degree 4
2( ) 2 7P x x x x a
where 2 or 7a
d) a polynomial of degree 5.
c) A monic polynomial of degree 4
2( ) 2 7P x x x x a
where 2 or 7a
d) a polynomial of degree 5.
2( ) 2 7P x x x
c) A monic polynomial of degree 4
2( ) 2 7P x x x x a
where 2 or 7a
d) a polynomial of degree 5.
2( ) 2 7P x x x ( )Q x
where - Q(x) is a polynomial of degree 2, and does not have a zero at 2 or –7
c) A monic polynomial of degree 4
2( ) 2 7P x x x x a
where 2 or 7a
d) a polynomial of degree 5.
2( ) 2 7P x x x ( )Q x
where - Q(x) is a polynomial of degree 2, and does not have a zero at 2 or –7
k
- k is a real number
4. If P(x) has degree n and has more than n real zeros, then P(x) is the zero polynomial. i.e. P(x) = 0 for all values of x
c) A monic polynomial of degree 4
2( ) 2 7P x x x x a
where 2 or 7a
d) a polynomial of degree 5.
2( ) 2 7P x x x ( )Q x
where - Q(x) is a polynomial of degree 2, and does not have a zero at 2 or –7
k
- k is a real number
Exercise 4E; 1, 3bd, 4a, 5b, 6a, 8, 12ad