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Factor each polynomial. $$(x+3)^{2}-2(x+3)+1$$

Describe some of the difficulties in factoring polynomials. What suggestions can you offer to overcome these difficulties?

You are about to take a great picture of fog rolling into San Francisco from the middle of the Golden Gate Bridge, 400 feet above the water. Whoops! You accidently lean too far over the safety rail and drop your camera. The height, in feet, of the camera after \(t\) seconds is modeled by the polynomial \(400-16 t^{2} .\) The factored form of the polynomial is \(16(5+t)(5-t) .\) Describe something about your falling camera that is easier to see from the factored form, \(16(5+t)(5-t),\) than from the form \(400-16 t^{2}\)

Determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. It takes a great deal of practice to get good at factoring a wide variety of polynomials.

Determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. Multiplying polynomials is relatively mechanical, but factoring often requires a great deal of thought.

Use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$4 x^{2}-9=(4 x+3)(4 x-3)$$

Determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. The factorable trinomial \(4 x^{2}+8 x+3\) and the prime trinomial \(4 x^{2}+8 x+1\) are in the form \(a x^{2}+b x+c\) but \(b^{2}-4 a c\) is a perfect square only in the case of the factorable trinomial.

Use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$x^{2}-6 x+9=(x-3)^{2}$$

Use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$4 x^{2}-4 x+1=(4 x-1)^{2}$$

Determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. When a factorization requires two factoring techniques, I'm less likely to make errors if I show one technique at a time rather than combining the two factorizations into one step.