Fundamental Theorem of Algebra

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Pre-Calculus › Fundamental Theorem of Algebra

Questions 1 - 10

Factorize the following polynomial expression completely to its linear factors: $f(x)=10x^{2}-4x+15xy-6y$

$(2x+3y)\cdot (5x-2)$

$(2x+3y\cdot (5x+2)$

$(2x-3y\cdot (5x+2)$

$(2x+3)\cdot (5x-2)$

Explanation

Use the grouping method to factorize common terms: $10x^{2}-4x+15xy-6y\rightarrow 2x(5x-2)\rightarrow (2x+3y)\cdot (5x-2)$

Factorize the following polynomial expression completely to its linear factors: $f(x)=10x^{2}-4x+15xy-6y$

$(2x+3y)\cdot (5x-2)$

$(2x+3y\cdot (5x+2)$

$(2x-3y\cdot (5x+2)$

$(2x+3)\cdot (5x-2)$

Explanation

Use the grouping method to factorize common terms: $10x^{2}-4x+15xy-6y\rightarrow 2x(5x-2)\rightarrow (2x+3y)\cdot (5x-2)$

Find a fourth-degree polynomial whose zeroes are , and $-2\pm \sqrt{2i}$

$x^{3}-16x^{2}-58x-60$

$x^{4}+x^{3}-16x^{2}-58x-60$

$x^{4}-x^{3}-16x^{2}-58x-60$

$x^{4}+x^{3}+16x^{2}+58x-60$

$x^{4}-16x^{2}-58x-60$

Explanation

This one is a bit of a journey. The expressions for the first two zeroes are easily calculated, $\left ( x+2 \right )$ and $\left ( x-5 \right )$ respectively. The last expression must be broken up into two equations: $x=-2-i\sqrt{2},x=-2+i\sqrt{2}$ which are then set equal to zero to yield the expressions $\left ( x+2-i\sqrt{2} \right )$ and $\left ( x+2+i\sqrt{2} \right )$

Finally, we multiply together all of the parenthesized expressions, which multiplies out to $x^{4}+x^{3}-16x^{2}-58x-60=0$

Find a fourth-degree polynomial whose zeroes are , and $-2\pm \sqrt{2i}$

$x^{3}-16x^{2}-58x-60$

$x^{4}+x^{3}-16x^{2}-58x-60$

$x^{4}-x^{3}-16x^{2}-58x-60$

$x^{4}+x^{3}+16x^{2}+58x-60$

$x^{4}-16x^{2}-58x-60$

Explanation

Finally, we multiply together all of the parenthesized expressions, which multiplies out to $x^{4}+x^{3}-16x^{2}-58x-60=0$

Factorize the following expression completely to its linear factors: f(x)=

$(x+5)\cdot(2x+y)$

$(x-5)\cdot(x+y)$

$(x-5)\cdot(2x-y)$

$(x-5)\cdot(2x+3y)$

$(x+5)\cdot(2x+y)$

Explanation

Use grouping method to factorize common terms:

$2x^2+3xy-10x-15y\rightarrow x(2x+3y)-5(2x+3y)\rightarrow (x-5)\cdot(2x+3y)$

Factorize the following expression completely to its linear factors: f(x)=

$(x+5)\cdot(2x+y)$

$(x-5)\cdot(x+y)$

$(x-5)\cdot(2x-y)$

$(x-5)\cdot(2x+3y)$

$(x+5)\cdot(2x+y)$

Explanation

Use grouping method to factorize common terms:

$2x^2+3xy-10x-15y\rightarrow x(2x+3y)-5(2x+3y)\rightarrow (x-5)\cdot(2x+3y)$

Express the polynomial as a product of linear factors:

Explanation

First pull out the common factor of 2, and then factorize:

Express the polynomial as a product of linear factors:

Explanation

First pull out the common factor of 2, and then factorize:

A polyomial with leading term $x^ {3}$ has 5 and 7 as roots; 7 is a double root. What is this polynomial?

$x^{3} - 19x^{2} + 119x -245$

$x^{3} -12x^{2} + 35x$

$x^{3}+ 19x^{2} +119x +245$

$x^{3}+ 17x^{2} +95x +175$

$x^{3}- 17x^{2} +95x -175$

Explanation

Since 5 is a single root and 7 is a double root, and the degree of the polynomial is 3, the polynomial is $(x-5)(x-7)^{2}$ . To put this in expanded form:

$(x-7)^{2} = x^{2} - 2 \cdotx\cdot 7 + 7^{2} = x^{2} - 14x + 49$

$(x-5)(x-7)^{2} =(x-5) \left ( x^{2} - 14x + 49 \right )$

$=x \left ( x^{2} - 14x + 49 \right ) -5\left( x^{2} - 14x + 4 9 \right )$

$= x^{3} - 14x^{2} + 49x - \left( 5x^{2} - 70x + 245 \right )$

$= x^{3} - 14x^{2} + 49x - 5x^{2} + 70x -245$

$= x^{3} - 19x^{2} + 119x -245$

A polyomial with leading term $x^ {3}$ has 5 and 7 as roots; 7 is a double root. What is this polynomial?

$x^{3} - 19x^{2} + 119x -245$

$x^{3} -12x^{2} + 35x$

$x^{3}+ 19x^{2} +119x +245$

$x^{3}+ 17x^{2} +95x +175$

$x^{3}- 17x^{2} +95x -175$

Explanation

Since 5 is a single root and 7 is a double root, and the degree of the polynomial is 3, the polynomial is $(x-5)(x-7)^{2}$ . To put this in expanded form:

$(x-7)^{2} = x^{2} - 2 \cdotx\cdot 7 + 7^{2} = x^{2} - 14x + 49$

$(x-5)(x-7)^{2} =(x-5) \left ( x^{2} - 14x + 49 \right )$

$=x \left ( x^{2} - 14x + 49 \right ) -5\left( x^{2} - 14x + 4 9 \right )$

$= x^{3} - 14x^{2} + 49x - \left( 5x^{2} - 70x + 245 \right )$

$= x^{3} - 14x^{2} + 49x - 5x^{2} + 70x -245$

$= x^{3} - 19x^{2} + 119x -245$

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