How do you simplify the expression $$\frac{2}{x-1} + \frac{4}{x+1}$$ ?

The simplified expression is $$\frac{6x-2}{(x-1)(x+1)}$$ .

How do you solve the equation $$\left(x^2-8\right)^2 + 10\left(x^2-8\right) + 21 = 0$$ ?

Let $$y = x^2 - 8$$ . The equation becomes $$y^2 + 10y + 21 = 0$$ , which factors to $$(y + 3)(y + 7) = 0$$ , giving solutions $$y = -3, -7$$ . Thus, $$x^2 = 5$$ leads to $$x = \pm\sqrt{5}$$ .

What is the first step in solving the equation $$6\left(x^2-1\right)^2 - \left(x^2-1\right) - 2 = 0$$ ?

Let $$z = x^2 - 1$$ . The equation becomes $$6z^2 - z - 2 = 0$$ .

What is the significance of the discriminant in a quadratic equation?

The discriminant, given by $$b^2 - 4ac$$ , determines the nature of the roots: if positive, there are two distinct real roots; if zero, one real root; if negative, no real roots.

What is the method for adding fractions with polynomials in the denominator?

To add fractions with polynomial denominators, factor the denominators, find the least common denominator (LCD), rewrite each fraction with the LCD, and then combine the numerators.

What is the result of simplifying $$\frac{7x}{x^2-5x} + \frac{x^2}{x-5}$$ ?

The simplified expression is $$\frac{x^2 + 7}{x - 5}; x \ne 0, 5$$ .

What is the importance of checking for extraneous solutions in equations?

Checking for extraneous solutions is important because some solutions derived from algebraic manipulations may not satisfy the original equation.

What is the process for factoring a quadratic expression?

To factor a quadratic expression $$ax^2 + bx + c$$ , find two numbers that multiply to $$ac$$ and add to $$b$$ , then rewrite the expression using these numbers.

How do you determine the vertical asymptotes of a rational function?

Vertical asymptotes occur at the values of $$x$$ that make the denominator zero, provided these values do not also make the numerator zero.

What is the relationship between the roots of a polynomial and its factors?

The roots of a polynomial are the values of $$x$$ that make the polynomial equal to zero, and each root corresponds to a factor of the polynomial.

What is the significance of the leading coefficient in a polynomial?

The leading coefficient determines the end behavior of the polynomial function; if positive, the function rises to the right, and if negative, it falls to the right.

What is the process for solving a polynomial equation?

To solve a polynomial equation, set the polynomial equal to zero, factor if possible, and then use the zero product property to find the roots.

ch 4 review flashcardizz #2 Flashcards

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1.

FLASHCARD QUESTION

Front

What is the process to add rational expressions with different denominators?

Back

To add rational expressions with different denominators, find a common denominator, rewrite each expression with that denominator, and then combine the numerators.

2.

FLASHCARD QUESTION

Front

What is the formula for factoring a difference of squares?

Back

3.

FLASHCARD QUESTION

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Back

4.

FLASHCARD QUESTION

Front

What is the quadratic formula used for?

Back

5.

FLASHCARD QUESTION

Front

What does it mean for a rational expression to be undefined?

Back

A rational expression is undefined when its denominator equals zero.

6.

FLASHCARD QUESTION

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Back

7.

FLASHCARD QUESTION

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Back

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