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Q1(ii):
Find the value of the polynomial $5x β 4x^2 + 3$ at
(ii) $x = β1$
Solution :
Initial Setup & Polynomial Definition
We are tasked with evaluating a given polynomial at a specific value of its variable. Let the given polynomial be denoted as $P(x)$.
The polynomial is defined as:
$P(x) = 5x - 4x^2 + 3$
For rigorous mathematical analysis, it is standard practice to rewrite the polynomial in descending order of its degree [Per the standard form of a polynomial $ax^2 + bx + c$]:
$P(x) = -4x^2 + 5x + 3$
Step 1: Substitution of the Variable
We must find the value of the polynomial at $x = -1$. This requires substituting $-1$ for every instance of the variable $x$ within the function $P(x)$.
$P(-1) = -4(-1)^2 + 5(-1) + 3$
Step 2: Evaluating the Exponent
According to the fundamental order of operations [PEMDAS/BODMAS], we must evaluate the exponent before performing multiplication.
The square of a negative number is positive [Since $(-a) \times (-a) = a^2$]:
$(-1)^2 = (-1) \times (-1) = 1$
Substituting this back into our equation yields:
$P(-1) = -4(1) + 5(-1) + 3$
Step 3: Executing the Multiplication
Next, we perform the scalar multiplications for each term:
- First term: $-4 \times 1 = -4$
- Second term: $5 \times (-1) = -5$
Substituting these products back into the polynomial expression:
$P(-1) = -4 - 5 + 3$
Step 4: Final Arithmetic Simplification
Finally, we perform addition and subtraction from left to right to find the scalar value of the polynomial.
Combine the negative terms:
$-4 - 5 = -9$
Add the constant term:
$P(-1) = -9 + 3$
$P(-1) = -6$
Graphical Verification
To provide a comprehensive understanding, the polynomial $P(x) = -4x^2 + 5x + 3$ represents a downward-opening parabola [Because the leading coefficient $a = -4$ is less than zero]. Evaluating the polynomial at $x = -1$ corresponds to finding the $y$-coordinate of the point on this curve where the $x$-coordinate is $-1$. As calculated, this point exists exactly at $(-1, -6)$.
Final Solution: The value of the polynomial $5x - 4x^2 + 3$ at $x = -1$ is $-6$.
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.2
- Q1(i): Find the value of the polynomial $5x β 4x^2 + 3$ at (i) $x = 0$
- Q1(iii): Find the value of the polynomial $5x β 4x^2 + 3$ at (iii) $x = 2$
- Q2(i): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (i) $p(y) = y^2 β y + 1$
- Q2(ii): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (ii) $p(t) = 2 + t + 2t^2 β t^3$
- Q2(iii): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (iii) $p(x) = x^3$
- Q2(iv): Find $p(0)$, $p(1)$ and $p(2)$ for each of the following polynomials: (iv) $p(x) = (x β 1) (x + 1)$
- Q3(i): Verify whether the following are zeroes of the polynomial, indicated against them. (i) $p(x) = 3x + 1, x = β\frac{1}{3}$
- Q3(ii): Verify whether the following are zeroes of the polynomial, indicated against them. (ii) $p(x) = 5x β \pi, x = \frac{4}{5}$
- Q3(iii): Verify whether the following are zeroes of the polynomial, indicated against them. (iii) $p(x) = x^2 β 1, x = 1, β1$
- Q3(iv): Verify whether the following are zeroes of the polynomial, indicated against them. (iv) $p(x) = (x + 1) (x β 2), x = β 1, 2$
- Q3(v): Verify whether the following are zeroes of the polynomial, indicated against them. (v) $p(x) = x^2, x = 0$
- Q3(vi): Verify whether the following are zeroes of the polynomial, indicated against them. (vi) $p(x) = lx + m, x = β\frac{m}{l}$
- Q3(vii): Verify whether the following are zeroes of the polynomial, indicated against them. (vii) $p(x) = 3x^2 β 1, x = β\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}$
- Q3(viii): Verify whether the following are zeroes of the polynomial, indicated against them. (viii) $p(x) = 2x + 1, x = \frac{1}{2}$
- Q4(i): Find the zero of the polynomial in each of the following cases: (i) $p(x) = x + 5$
- Q4(ii): Find the zero of the polynomial in each of the following cases: (ii) $p(x) = x β 5$
- Q4(iii): Find the zero of the polynomial in each of the following cases: (iii) $p(x) = 2x + 5$
- Q4(iv): Find the zero of the polynomial in each of the following cases: (iv) $p(x) = 3x β 2$
- Q4(v): Find the zero of the polynomial in each of the following cases: (v) $p(x) = 3x$
- Q4(vi): Find the zero of the polynomial in each of the following cases: (vi) $p(x) = ax, a \neq 0$
- Q4(vii): Find the zero of the polynomial in each of the following cases: (vii) $p(x) = cx + d, c \neq 0, c, d$ are real numbers.
CBSE Solutions for Class 9 Mathematics Polynomials
Chapters in CBSE - Class 9 Mathematics
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