(D) For a quadratic polynomial of the form $ax^2 + bx + c$,the sum of the zeros is given by $-\frac{b}{a}$ and the product of the zeros is given by $\

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(D) For a quadratic polynomial of the form $ax^2 + bx + c$,the sum of the zeros is given by $-\frac{b}{a}$ and the product of the zeros is given by $\frac{c}{a}$.Given the polynomial $3x^2 + x - 4$,we have $a = 3$,$b = 1$,and $c = -4$.Sum of zeros $= -\frac{b}{a} = -\frac{1}{3}$.Product of zeros $= \frac{c}{a} = \frac{-4}{3} = -\frac{4}{3}$.Thus,the sum and product are $-\frac{1}{3}$ and $-\frac{4}{3}$ respectively.

Class 10 Mathematics Polynomials Mix Examples - Polynomials

Medium

Obtain the sum and the product of the zeros of the following quadratic polynomial without finding the zeros: $3x^2 + x - 4$.

A
$-\frac{7}{3}, -\frac{5}{3}$
B
$-\frac{9}{3}, -\frac{9}{3}$
C
$-\frac{2}{3}, -\frac{4}{2}$
D
$-\frac{1}{3}, -\frac{4}{3}$

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Mix Examples - Polynomials

The graph of a linear polynomial $p(x) = ax + b$ intersects the $X$-axis at $\ldots \ldots \ldots \ldots$ (where $a \neq 0$).

Easy

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Verify that $3$ is a zero of the linear polynomial $p(x) = 4x - 12$.

Easy

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The zeros of cubic polynomial $p(x) = ax^3 + bx^2 + cx + d$; $a \neq 0, a, b, c, d \in R$ are $\alpha, \beta$ and $\gamma$; then $\alpha\beta + \beta\gamma + \gamma\alpha = \ldots$

Medium

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Divide the following by the synthetic division method: $p(x) = x^{4} - 1$ by $x - 1$.

Medium

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Which of the following groups matches the data of Part $I$ with the data of Part $II$?

Part $I$ Part $II$

$1.$ The graph of $p(x)=x^{2}-5x+6$ $a.$ Intersects the $X$-axis at one point.

$2.$ The graph of $p(x)=x^{2}-10x+25$ $b.$ Intersects the $X$-axis at zero points.

$3.$ The graph of $p(x)=x^{3}-4x$ $c.$ Intersects the $X$-axis at two points.

$4.$ The graph of $p(x)=x^{2}+4x+5$ $d.$ Intersects the $X$-axis at three points.

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Part $I$	Part $II$
$1.$ The graph of $p(x)=x^{2}-5x+6$	$a.$ Intersects the $X$-axis at one point.
$2.$ The graph of $p(x)=x^{2}-10x+25$	$b.$ Intersects the $X$-axis at zero points.
$3.$ The graph of $p(x)=x^{3}-4x$	$c.$ Intersects the $X$-axis at two points.
$4.$ The graph of $p(x)=x^{2}+4x+5$	$d.$ Intersects the $X$-axis at three points.

Obtain the sum and the product of the zeros of the following quadratic polynomial without finding the zeros: $3x^2 + x - 4$.

Similar Questions

The graph of a linear polynomial $p(x) = ax + b$ intersects the $X$-axis at $\ldots \ldots \ldots \ldots$ (where $a \neq 0$).

Verify that $3$ is a zero of the linear polynomial $p(x) = 4x - 12$.

The zeros of cubic polynomial $p(x) = ax^3 + bx^2 + cx + d$; $a \neq 0, a, b, c, d \in R$ are $\alpha, \beta$ and $\gamma$; then $\alpha\beta + \beta\gamma + \gamma\alpha = \ldots$

Divide the following by the synthetic division method: $p(x) = x^{4} - 1$ by $x - 1$.

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