Module 4 Homework Assignment 1 Math 110 B

Module 4 Homework Assignment 1mat 110 B

Find the greatest common factor of the numbers 4, 6, and 12. Factor the expression 24x^3 + 30x, including factoring out the GCF with a negative coefficient. Factor completely by factoring out any common factors and then factoring by grouping the expression 6x^2 – 5xy + 6x – 5y. Find the GCF of the binomials (15y + 20) and (15y + 21) and then determine the GCF of their product.

The area of a rectangle with length x is given by 15x – x^2. Find the width of the rectangle in terms of x. Factor the trinomial x^2 + 8x – some constant, as well as 2x^2 + 16x + some constant. Complete the statement: 6a^2 – 5a + 1 = (3a – 1)(__ ? __). Determine whether the following quadratic equations are true or false: x^2 – 7x – 30 = (x + 3)(x – some constant). Factor the quadratic expressions x^2 + 11x + some constant and 15x^2 + 23x + some constant. Factor completely the expression 6z^3 – 27z^2 + 12z.

The number of hot dogs sold at each hour h after opening at a soccer tournament is given by the polynomial 2h^2 – 19h + 24. Write this polynomial in factored form. Find a positive value of k for which the polynomial x^2 – kx + some constant can be factored, and then factor completely the quadratic 9x^2 + some constant. Determine whether a given quadratic trinomial is a perfect square, and if so, factor it.

Factor the expression x^2 – 12x + some constant and 25x^2 + 40xy + 16y. Factor the expression s^2(t – u) – 9t^2(t – u) and identify the appropriate method to factor the polynomial 6x^3 + 9x. Similarly, identify the first step for factoring 2a^2 + 9a + some constant and solve the quadratic equations 5x^2 + 17x = – and 3x(2x – 15) = –.

The sum of an integer and its square equals 30; find the integer. If the sides of a square are decreased by 3 cm, the area decreases by 81cm^2; find the original dimensions. Write algebraic expressions in simplest form, perform multiplication, division, addition, and subtraction, and find the area of a rectangle given by certain expressions. Simplify algebraic fractions and perform operations among algebraic expressions.

Determine which values for x must be excluded in certain algebraic fractions, such as those involving factors in denominators. Solve for x in equations like 5x + 6 = some value, and 3 + 5 – 3 – 3x = some value. Find numbers related by ratios and reciprocals, like one being three times another, and solve for specific variables.

Paper For Above instruction

Algebraic operations and polynomial factorizations are foundational skills in algebra that enable the simplification and solution of various equations. This paper explores these concepts through a series of problems, illustrating methods such as finding the greatest common factor (GCF), factoring polynomials, solving quadratic equations, and applying algebra in real-world contexts such as areas and ratios.

Understanding Greatest Common Factors

The GCF is the largest factor that divides two or more numbers or expressions without leaving a remainder. For example, finding the GCF of 4, 6, and 12 involves listing their factors: 4 (1, 2, 4), 6 (1, 2, 3, 6), and 12 (1, 2, 3, 4, 6, 12). The common factors are 1, 2, and the greatest is 2. Therefore, the GCF of the three numbers is 2.

In polynomial expressions, the GCF is found by identifying the common factors of each term's coefficients and variables. For instance, in 24x^3 + 30x, the GCF of 24 and 30 is 6, and each term contains at least one x, so the GCF of the whole expression is 6x. Factoring out 6x yields 6x(4x^2 + 5).

Factoring Techniques

Factoring quadratic expressions is a vital skill. For quadratic trinomials like x^2 + 8x + c, where c is a constant, factoring involves finding two numbers that multiply to c and add to 8. If the trinomial is x^2 + 8x + 15, then factors of 15 are 3 and 5, which sum to 8, leading to (x + 3)(x + 5).

Complete the square or use quadratic formulas when applicable, especially for quadratics where factoring directly isn't straightforward. For example, the quadratic x^2 – 7x – 30 factors into (x + 3)(x – 10), as 3 and –10 multiply to –30 and add to –7.

Polynomial Operations and Applications

Multiplying binomials utilizes the FOIL method, adding the products of the outer, inner, and last terms. Dividing polynomials involves polynomial long division or synthetic division when dividing by linear factors. These techniques facilitate simplifying complex algebraic expressions and solving equations efficiently.

In real-world applications, such as calculating area, expressing it as a quadratic formula allows for solving for dimensions. For instance, if the area of a rectangle is given by 15x – x^2, then solving for width involves expressing the area in vertex form or factored form. Similarly, problems involving shadows and similar triangles provide practical contexts for applying ratios and proportions, e.g., the height of a tree based on shadow lengths and the pole's height.

Quadratic Equations and Their Solutions

Quadratic equations are solved via factoring, completing the square, or quadratic formula. For example, solving 5x^2 + 17x = – involves setting the equation to zero and factoring or applying the quadratic formula to find solutions for x. When quadratic expressions are perfect squares, they can be factored into binomials involving squares, e.g., x^2 – 12x + 36 = (x – 6)^2.

Moreover, determining extraneous solutions requires understanding domains, especially where denominators involve variables, such as in algebraic fractions. Values that make denominators zero must be excluded, ensuring solutions are valid within the domain of the expression.

Algebra in Word Problems and Ratios

Setting up algebraic models for word problems involves translating statements into equations. For example, one number being 8 less than another can be written as x and x – 8, then combined to find their reciprocals' sum. In ratios, if one quantity is three times another, variables can be assigned accordingly, and equations can be solved systematically.

In problems involving shadows, similar triangles are used to establish ratios, such as height to shadow length, enabling calculation of unknown heights. These applications underscore how algebraic concepts are vital for solving practical problems in fields like physics, engineering, and everyday life.

Conclusion

Mastering the techniques of factoring, solving quadratic equations, simplifying expressions, and applying algebra to word problems provides essential skills for progressing in mathematics. These skills facilitate understanding complex problems, making calculations manageable, and applying mathematical reasoning to real-world situations. As such, continued practice and application of these concepts are crucial for developing proficiency in algebra.

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