(A) quadratic polynomial is given by the formula: $p(x) = k[x^2 - (\text{sum of zeros})x + (\text{product of zeros})]$.Given:Sum of zeros $= \frac{1}{

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(A) quadratic polynomial is given by the formula: $p(x) = k[x^2 - (\text{sum of zeros})x + (\text{product of zeros})]$.Given:Sum of zeros $= \frac{1}{4}$Product of zeros $= -1$Substituting these values into the formula:$p(x) = k[x^2 - (\frac{1}{4})x + (-1)]$$p(x) = k[x^2 - \frac{1}{4}x - 1]$To simplify,we can take $k = 4$:$p(x) = 4[x^2 - \frac{1}{4}x - 1] = 4x^2 - x - 4$.Thus,the polynomial is $k(4x^2 - x - 4)$.

Class 10 Mathematics Polynomials Mix Examples - Polynomials

Easy

Obtain a quadratic polynomial with the following conditions:
The sum of the zeros $= \frac{1}{4}$;
The product of the zeros $= -1$.

A
$k(4x^2 - x - 4)$
B
$k(4x^2 + x - 4)$
C
$k(x^2 - \frac{1}{4}x - 1)$
D
$k(4x^2 - x + 4)$

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Mix Examples - Polynomials

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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Divide $3 x^{4}+5 x^{3}-7 x^{2}+2 x+2$ by $x^{2}+3 x+1$.

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If $\alpha, \beta$ and $\gamma$ are the zeros of the cubic polynomial $p(x) = x^{3} - 6x^{2} - 7x$,then $\alpha + \beta + \gamma = \ldots$

Medium

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The value of $p(x) = 4x^3 + 3x^2 + 2x + 1$ at $x = 1$ is $\ldots \ldots \ldots$

Easy

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What must be added to the polynomial $p(x) = 6x^5 + 5x^4 + 11x^3 - 3x^2 + x + 1$ so that the resulting polynomial is exactly divisible by $3x^2 - 2x + 4$? (Hint: add $-r(x)$ to $p(x)$)

Medium

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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 a quadratic polynomial with the following conditions:The sum of the zeros $= \frac{1}{4}$;The product of the zeros $= -1$.

Similar Questions

Divide $3 x^{4}+5 x^{3}-7 x^{2}+2 x+2$ by $x^{2}+3 x+1$.

If $\alpha, \beta$ and $\gamma$ are the zeros of the cubic polynomial $p(x) = x^{3} - 6x^{2} - 7x$,then $\alpha + \beta + \gamma = \ldots$

The value of $p(x) = 4x^3 + 3x^2 + 2x + 1$ at $x = 1$ is $\ldots \ldots \ldots$

What must be added to the polynomial $p(x) = 6x^5 + 5x^4 + 11x^3 - 3x^2 + x + 1$ so that the resulting polynomial is exactly divisible by $3x^2 - 2x + 4$? (Hint: add $-r(x)$ to $p(x)$)

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Obtain a quadratic polynomial with the following conditions:
The sum of the zeros $= \frac{1}{4}$;
The product of the zeros $= -1$.