Determine the exact value of k and m such for which the polynomial function p(x) = x^4 + x^3 + kx^2 + mx − 15 has all the following properties:
• the graph of p(x) crosses the x-axis at x = 1
• division by x + 1 gives a remainder of 2
Answers
Given polynomial is
Now, given that,
The graph of p(x) crosses the x-axis at x = 1.
It means, x = 1 is zero of p(x).
It means,
Also, given that
When p(x) is divided by x + 1, it gives a remainder of 2.
We know,
Remainder Theorem states that when a polynomial f(x) is divided by linear polynomial x - a, then remainder is f(a).
So,
On adding equation (2) and (3), we get
On substituting the value of k in equation (3), we get
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More to know :-
ALGEBRAIC IDENTITIES
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b² = (a + b)(a - b)
(a + b)² = (a - b)² + 4ab
(a - b)² = (a + b)² - 4ab
(a + b)² + (a - b)² = 2(a² + b²)
(a + b)³ = a³ + b³ + 3ab(a + b)
(a - b)³ = a³ - b³ - 3ab(a - b)
Given polynomial is
Now, given that,
The graph of p(x) crosses the x-axis at x = 1.
It means, x = 1 is zero of p(x).
It means,
Also, given that
When p(x) is divided by x + 1, it gives a remainder of 2.
We know,
Remainder Theorem states that when a polynomial f(x) is divided by linear polynomial x - a, then remainder is f(a).
So,
On adding equation (2) and (3), we get
On substituting the value of k in equation (3), we get
▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬
More to know :-
ALGEBRAIC IDENTITIES
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
a² - b² = (a + b)(a - b)
(a + b)² = (a - b)² + 4ab
(a - b)² = (a + b)² - 4ab
(a + b)² + (a - b)² = 2(a² + b²)
(a + b)³ = a³ + b³ + 3ab(a + b)
(a - b)³ = a³ - b³ - 3ab(a - b)