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This is one of our many quizzes on Polynomials in standard form. Click on this to access more such competitive quizzes on Polynomials.

Polynomials in standard form Quiz 2

mathematics

 The standard form of  $8x - {x^2} = 57 + 3x - 1$ is:

 After the simplification, arrange the terms of the right side of the equations in descending order according to the exponent values of the variable. 

Given, $8x - {x^2} = 57 + 3x - 1$. $ \\ \Rightarrow 8x - {x^2} - 57 - 3x + 1 = 0 \\   \Rightarrow 5x - {x^2} - 56 = 0 \\   \Rightarrow  - {x^2} + 5x - 56 = 0 \\  $

 The degree of the polynomial ${a^3}\left( {{a^2} + a + 1} \right)$ is:

 Multiply the term outside the bracket with each term inside the bracket and then determine the highest exponent value.

Given the polynomial ${a^3}\left( {{a^2} + a + 1} \right)$. $ \\ $ To determine its degree, we have to solve the bracket first. $ \\ {a^3}\left( {{a^2} + a + 1} \right) \\   = {a^{3 + 2}} + {a^{3 + 1}} + {a^3} \\   = {a^5} + {a^4} + {a^3} \\  $ Therefore the highest power is of the first term and hence the degree of the given polynomial is 5.

 Determine the degree of the polynomial $7a - 3bc$..

 Whenever two variables are given in the single term, add their exponent values to determine the degree of that term.

We are given the polynomial $7a - 3bc$. $ \\ $ The power of the first term is 1. $ \\ $ Second term, two variables are given, each with power 1.  $ \\ $ Therefore, the power or the degree of the second term will be $1 + 1 = 2$. $ \\ $ Thus, the power of the second term is more than that of the first term.  $ \\ $ Hence, the degree of the given polynomial is 2.

 The standard form of a quadratic equation is $a{x^2} + bx + c = 0$. If it has two roots $\dfrac{1}{2}$ and $5$, what will be the values of $a$, $b$ and $c$ respectively?

 Determine the factors from the roots and then multiply them to get the standard form of the quadratic equation.

It is given that the quadratic equation has two roots $\dfrac{1}{2}$and $5$. $ \\ x = \dfrac{1}{2} \\   \Rightarrow 2x = 1 \\   \Rightarrow 2x - 1 = 0 \\  $ and $  x = 5 \\   \Rightarrow x - 5 = 0 \\  $ Thus, we can say that the quadratic equation has two factors $\left( {2x - 1} \right)$ and $\left( {x - 5} \right)$. $ \\ $ The equation can be written as the multiplication of these factors. $ \\  \Rightarrow \left( {2x - 1} \right)\left( {x - 5} \right) = 0 \\   \Rightarrow 2{x^2} - 10x - x + 5 = 0 \\   \Rightarrow 2{x^2} - 11x + 5 = 0 \\  $ Now if we compare this equation with $a{x^2} + bx + c = 0$, we will get $a = 2$, $b =  - 11$ and $c = 5$.

 Consider a line passing through the point $\left( {4,4} \right)$ and has a slope of $ - 2$. What is the standard form of the equation of this line?

 Use the standard equation of line and the given information to find the y-intercept and then put this value to get the required answer.

 W.K.T the standard form of the equation of line is $y = mx + c$. $ \\ $ Given a point through which the line is passing and the value of the slope. $ \\ \left( {x,y} \right) = \left( {4,4} \right)$ and $m =  - 2 \\ $ Now, use these values to get the y-intercept of the line. $ \\ y = mx + c \\   \Rightarrow 4 = \left( { - 2 \times 4} \right) + c \\   \Rightarrow c = 4 + 8 = 12 \\  $ Thus, put the values of slope and y-intercept, to get the standard form of the equation of the given line as: $y =  - 2x + 12$